A STRAIGHTENING LAW FOR THE
DRINFEL’D LAGRANGIAN GRASSMANNIAN
A Dissertation
by
JAMES VINCENT RUFFO
Submitted to the Office of Graduate Studies of
Texas A&M University
in partial fulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHY
August 2007
Major Subject: Mathematics
A STRAIGHTENING LAW FOR THE
DRINFEL’D LAGRANGIAN GRASSMANNIAN
A Dissertation
by
JAMES VINCENT RUFFO
Submitted to the Office of Graduate Studies of
Texas A&M University
in partial fulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHY
Approved by:
Chair of Committee, Frank Sottile
Committee Members, John Keyser
J. Maurice Rojas
Henry Schenck
Head of Department, Al Boggess
August 2007
Major Subject: Mathematics
iii
ABSTRACT
A Straightening Law for the
Drinfel’d Lagrangian Grassmannian. (August 2007)
James Vincent Ruffo, B.A.;B.S., University of Rochester;
M.A., University of Massachusetts-Amherst
Chair of Advisory Committee: Dr. Frank Sottile
The Drinfel’d Lagrangian Grassmannian compactifies the space of algebraic
maps of fixed degree from the projective line into the Lagrangian Grassmannian. It
has a natural projective embedding arising from the highest weight embedding of
the ordinary Lagrangian Grassmannian, and one may study its defining ideal in this
embedding. The Drinfel’d Lagrangian Grassmannian is singular. However, a concrete
description of generators for the defining ideal of the Schubert subvarieties of the
Drinfel’d Lagrangian Grassmannian would imply that the singularities are modest.
I prove that the defining ideal of any Schubert subvariety is generated by polynomials which give a straightening law on an ordered set. Using this fact, I show that
any such subvariety is Cohen-Macaulay and Koszul. These results represent a partial
extension of standard monomial theory to the Drinfel’d Lagrangian Grassmannian.
iv
ACKNOWLEDGMENTS
I thank my advisor Frank Sottile for introducing me to this topic, and for his support and guidance. I also thank Dimitrije Kostić, J.M. Landsberg, Mauricio Velasco
and Alexander Woo for helpful conversations and suggestions.
v
TABLE OF CONTENTS
CHAPTER
Page
I
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . .
1
II
BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . .
5
A. Basic theory . . . . . . . . . . . . . . . . . . . . .
1. Algebraic geometry . . . . . . . . . . . . . .
2. Representation theory . . . . . . . . . . . . .
B. Flag varieties . . . . . . . . . . . . . . . . . . . .
1. The Lagrangian Grassmannian . . . . . . . .
C. Spaces of algebraic maps . . . . . . . . . . . . . .
D. Algebras with straightening laws . . . . . . . . . .
1. Hilbert series of an algebra with straightening
2. The doset of admissible pairs . . . . . . . . .
III
.
.
.
.
.
.
.
.
.
5
5
10
16
16
26
28
31
36
RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
A. Algebras with straightening law . . . . . . . . .
B. Drinfel’d flag varieties . . . . . . . . . . . . . .
1. A basis for Sd C2 ⊗ L(ωn )∗ . . . . . . . . . .
2. Proof and consequences of the straightening
3. Representation-theoretic interpretation . . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
law
. . .
.
.
.
.
.
40
42
42
50
56
SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
IV
. . .
. . .
. . .
law
. . .
.
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vi
LIST OF FIGURES
FIGURE
Page
1
The partition (3, 3, 1) associated to 4̄2̄23.
. . . . . . . . . . . . . . .
20
2
The paths associated to 4̄2̄13 and 3̄1̄24 in P4 . . . . . . . . . . . . . .
23
3
Elements of πn−1 (4̄2̄13, 3̄1̄24).
. . . . . . . . . . . . . . . . . . . . . .
24
4
The set Ch(D) of chains in D. . . . . . . . . . . . . . . . . . . . . . .
32
5
The doset of Example 2.50 . . . . . . . . . . . . . . . . . . . . . . . .
36
6
The doset D2,4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
7
The subset of S2,4 associated to 4̄2̄13(1) ∈ P2,4 . . . . . . . . . . . . . .
39
8
The two cases in the proof of fourth condition in Definition 2.45. . .
52
9
The module S 2 (S d C2 ⊗L(ω)) decomposes into factors corresponding to the colored boxes. . . . . . . . . . . . . . . . . . . . . . . . . .
59
10
A Young scheme on λ/µ(−1) . . . . . . . . . . . . . . . . . . . . . .
61
11
A Young scheme on λ/µ . . . . . . . . . . . . . . . . . . . . . . . . .
61
1
CHAPTER I
INTRODUCTION
Enumerative geometry concerns problems of counting geometric objects incident
upon other objects in a prescribed way. In the second half of the 19th century, many
remarkable numbers in enumerative geometry were computed – for example, in 1864
Chasles proved that there are 3264 plane conics tangent to 5 fixed conics [20].
The Schubert calculus is concerned with an important subset of enumerative
geometry. It provides a framework for counting linear subspaces of a vector space
satisfying conditions imposed by other linear subspaces. We call a k-dimensional
linear subspace a k-plane, or if k = 1, a line. Note that we require that these
objects pass through the origin. The smallest non-trivial example of a problem in the
Schubert calculus is the following.
Question 1.1. Given four general fixed 2-planes in C4 , how many 2-planes meet all
of them?
It turns out that the answer is 2. The use of the term “general” is in the technical
sense of algebraic geometry. We think of the four 2-planes as being fixed in space,
and the problem is to count the number of ways to place a 2-plane in some position so
that it intersects each of the four 2-planes non-trivially. We thus refer to the objects
imposing the conditions as fixed.
More generally, one may ask:
Question 1.2. How many k-planes in n-dimensional space meet k(n−k) fixed general
n−k-planes non-trivially?
This dissertation follows the style of Representation Theory.
2
The answer, first computed by Schubert [37], is
1!2! · · · k!(k(n − k))!
.
(n − k)!(n−k+1)! · · · n!
The Schubert calculus deals not only with linear subspaces, but with nested chains
E• := (Ea1 ⊂ Ea2 ⊂ · · · ⊂ Eas ⊂ Cn ) with Eai an ai -plane in Cn , for each i = 1, . . . , s.
We say that E• is a flag of type (a1 , a2 , . . . , as ) in Cn . An example of a problem
involving flags is answered by the following proposition.
Proposition 1.3. There are 2 flags (E2 ⊂ E3 ) of type (2, 3) in 4-dimensional space
C4 such that E2 meets 3 general fixed 2-planes and E3 contains 2 fixed lines.
In the modern setting, these problems are solved by introducing a parameter
space for the objects to be counted, called a flag variety. This is a smooth projective variety, and it has subvarieties (called Schubert varieties) which parametrize
flags satisfying incidence conditions with respect to a fixed flag. (see Chapter II for
background material). The problem is thus reduced to the problem of computing the
number of points in an intersection of Schubert varieties. These points are the set of
solutions to a system of polynomial equations determined by the Schubert varieties.
Flag varieties are the compact homogeneous spaces for the action of a complex
semisimple algebraic group G. All of the examples above involve flag varieties associated to a special linear group. For us, the symplectic group Sp2n (C) will figure
prominently.
Example 1.4. Let Ω be a non-degenerate alternating bilinear form on the vector
space C4 . The Lagrangian Grassmannian X := LG(2) is the set of all 2-planes
E ⊂ C4 such that Ω is identically zero on E. Such subspaces are called isotropic.
The symplectic group G := Sp4 C is the group of invertible linear transformations
T : C4 → C4 such that Ω(T (v), T (w)) = Ω(v, w) for all v, w ∈ C4 . The group G acts
3
transitively on X. Pick a point x ∈ X, and let P ⊂ G be its stabilizer. There is
a bijection X ∼
= G/P . In the literature the terms “flag variety” and G/P are used
interchangeably.
One may pose enumerative questions on X, for example:
Question 1.5. How many isotropic 2-planes x ∈ X meet 3 general isotropic 2-planes?
Again, the answer is 2.
Recently there has been much interest in enumerative problems involving algebraic maps from a complete curve C to a flag variety. These problems involve
conditions that the image of such a map pass through a Schubert variety at some
point c ∈ C, and can be studied using the quantum cohomology ring of the flag
variety [14].
These problems can be posed as intersections on an auxiliary space parametrizing such algebraic maps. This space has applications to mathematical physics, linear
systems theory, geometric representation theory, and the geometric Langlands correspondence, as well as quantum cohomology (see [2, 39, 40] for details). It is (almost)
never compact, so various compactifications have been introduced to help understand
its geometry. Among these are Kontsevich’s space of stable maps [14, 21], the quot
scheme (or space of quasiflags) [6, 32, 42], and, at least when C is the projective line
P1 , the Drinfel’d compactification (or space of quasimaps) [2, 22]. This latter space
is defined concretely as a projective variety, and it is amenable to direct study via
its defining equations. In particular, one may apply some of the ideas of standard
monomial theory.
Inspired by the work of Hodge [19] on the Grassmannian, Lakshmibai, Musili,
Seshadri, and others (see [23, 35] and references therein) developed standard monomial theory to study the flag varieties G/Q, where G is a semisimple algebraic group
4
and Q ⊂ G is a parabolic subgroup. These spaces have a decomposition into Schubert cells, whose closures (the Schubert varieties) give a basis for cohomology. A
consequence of standard monomial theory is that Schubert varieties are normal and
Cohen-Macaulay, and one has an explicit description of their singularities and defining
ideals.
A key part of standard monomial theory is that any G/P (P a maximal parabolic
subgroup) has a projective embedding with coordinates such that the defining ideal of
G/P takes the form of a straightening law (Definition 2.45). This idea originates with
the work of Hodge on the Grassmannian [19], and was extended to the Lagrangian
Grassmannian by DeConcini and Lakshmibai [9]. An algebra with straightening law
is a special case of a Hodge algebra [5].
Sottile and Sturmfels have extended standard monomial theory to the Drinfel’d
Grassmannian parametrizing algebraic maps from P1 into the Grassmannian [41].
They extend the definition of Schubert varieties to this space, and prove that the
homogeneous coordinate ring of any Schubert variety (including the Drinfel’d Grassmannian itself) is an algebra with straightening law on a distributive lattice. This
is the key fact that allows one to prove that these Schubert varieties are normal,
Cohen-Macaulay, and Koszul, and have rational singularities.
We extend these results to the Drinfel’d Lagrangian Grassmannian, defined in
Chapter II, Section C. The necessary background on algebras with straightening law
is provided in Chapter II, Section D. Our main results are proved in Chapter III,
along with some consequences.
5
CHAPTER II
BACKGROUND
A. Basic theory
1. Algebraic geometry
We review some facts from algebraic geometry, which is the study of the solution
sets of systems of polynomial equations. We recall the basic theorems and definitions
of the subject, emphasizing those ideas which will be of particular interest to us.
See [11, 17, 18] for more details on the general theory and [10, 16] for more on
Gröbner bases.
Let C[x1 , . . . , xn ] denote the ring of polynomials in the variables x1 , . . . , xn with
complex coefficients. Let An := {a = (a1 , . . . , an ) | ai ∈ C, i = 1, . . . , n} denote
n-dimensional affine space over C. Associated to a subset S ⊂ C[x1 , . . . , xn ] we have
the affine variety
V(S) := {a ∈ An | f (a) = 0, for all f ∈ S} .
Conversely, to a subset X ⊂ An , we associate the set of polynomials vanishing on X
I(X) := {f ∈ C[x1 , . . . , xn ] | f (a) = 0, for all a ∈ X} .
These associations form the basis for a correspondence between algebraic and geometric objects. Several refinements must be made to make this correspondence precise.
First, note that for any subset X ⊂ An , I(X) is an ideal. That is, 0 ∈ I(X), and
f, g ∈ I(X) and b ∈ C[x1 , . . . , xn ] imply that f + bg ∈ I(X). Hence we may restrict
attention to ideals of C[x1 , . . . , xn ]. Hilbert’s Basis Theorem [11] implies that every
6
such ideal I is finitely generated, i.e., there exist f1 , . . . , fr ∈ C[x1 , . . . , xn ] such that
I = {g1 f1 + · · · + gr fr | g1 , . . . , gr ∈ C[x1 , . . . , xn ]}
We write I = hf1 , . . . , fr i in this situation.
For a given variety X ⊂ An , there are many ideals I such that V(I) = X, and
I(X) contains all of them. Ideals of the form I(X) can be characterized purely in
√
terms of algebra. The radical of I, denoted I, is the ideal of all f ∈ C[x1 , . . . , xn ]
√
such that f N ∈ I, for some N ∈ N. I is a radical ideal if I = I.
Theorem 2.1 (Hilbert’s Nullstellesatz). Let I ⊂ C[x1 , . . . , xn ] be an ideal. Then
√
I(V(I)) =
I.
We can now state one version of the algebro-geometric dictionary:
Proposition 2.2. The maps X 7→ I(X) and I 7→ V(I) give an inclusion reversing
bijection between radical ideals in the ring C[x1 , . . . , xn ] and affine varieties contained
in An .
In a similar fashion, one may consider subvarieties of projective space. Recall that
n-dimensional projective space Pn is the set of all 1-dimensional linear subspaces of the
n+1-dimensional vector space Cn+1 . Let x0 , . . . , xn be coordinate functions on Cn+1
∗
(that is, a basis for the dual vector space Cn+1 ). These are homogeneous coordinates
on Pn . A projective variety is the set of zeroes in Pn of some set of polynomials in
P
C[x0 , . . . , xn ]. A polynomial in C[x0 , . . . , xn ] can be written as f = N
i=1 fi , where
each fi is a homogeneous polynomial of degree i; the polynomial f vanishes on a point
of Pn if and only if each homogeneous component fi does so. Hence the defining ideal
of a projective variety is homogeneous, in that it is generated by a set of homogeneous
polynomials.
7
Note that the irrelevant ideal hx0 , . . . , xn i defines the empty set in Pn , as does
any homogeneous ideal containing some power hx0 , . . . , xn im (m ∈ N) of this ideal.
Moreover, a polynomial f ∈ C[x0 , . . . , xn ] vanishes on a given projective variety if
N
and only if for some N , all of the polynomials xN
0 f, . . . , xn f vanish. This motivates
the following definition.
Definition 2.3. The saturation of an ideal I ⊂ C[x0 , . . . , xn ] is the ideal
sat I = {f ∈ C[x0 , . . . , xn ] | xN
i f ⊂ I for all i = 0, . . . , n and some N ∈ N} .
The ideal I is saturated if sat I = I.
We again write V(I) for the zero set in Pn of a homogeneous ideal and I(X) for
the defining ideal of a projective variety. One has the following homogeneous version
of the algebro-geometric dictionary.
Proposition 2.4. The maps X 7→ I(X) and I 7→ V(I) give an inclusion reversing
bijection between saturated radical homogeneous ideals in the ring C[x0 , . . . , xn ] and
projective varieties in Pn .
Gröbner bases have driven a resurgence of computational methods in algebraic
geometry in the past two decades. We recall the basics.
We write xa for the monomial xa11 · · · xann ∈ C[x1 , . . . , xn ]. Let deg xa :=
be the total degree of xa .
Pn
i=1
ai
A term order < is a total ordering on the set of monomials in C[x1 , . . . , xn ]
such that 1 is the unique minimal element of C[x1 , . . . , xn ], and such that for any
monomials m1 , m2 , and m in C[x1 , . . . , xn ], m1 < m2 implies that mm1 < mm2 .
Consider the linear ordering of the variables given by x1 > x2 > . . . > xn . Some
of the most frequently used term orders are the following:
8
1. xa ≤ xb if and only if the first nonzero entry of b − a is positive. This defines
the lexicographic term order on C[x1 , . . . , xn ].
2. xa ≤ xb if and only if deg xa < deg xb , or deg xa = deg xb and the first nonzero
entry of b − a is positive. This defines the degree lexicographic term order on
C[x1 , . . . , xn ].
3. xa ≤ xb if and only if deg xa < deg xb , or deg xa = deg xb and the last nonzero
entry of b − a is negative. This defines the degree reverse lexicographic term
order on C[x1 , . . . , xn ].
Given a term order <, we define the initial monomial in< f of f to be the largest
monomial (with respect to <) appearing in f . For I ⊂ C[x1 , . . . , xn ] an ideal, the
initial ideal in< I is the ideal generated by the set {in< f | f ∈ I}.
Definition 2.5. Let I ⊂ C[x1 , . . . , xn ] be an ideal. Then a set {f1 , . . . , fk } ⊂ I is a
Gröbner basis for I if in< I is generated by {in< f1 , . . . , in< fk }.
Proposition 2.6. Let {f1 , . . . , fk } ⊂ I be a Gröbner basis for I. Then I is generated
by {f1 , . . . , fk }.
Example 2.7. Let f1 := x2 − y and f2 := x3 − z. Consider the ideal I := (f1 , f2 ) ⊂
C[x, y, z]. Let < denote the lexicographic ordering on C[x, y, z], so that the initial
terms are those that are underlined above. The polynomial
f3 := xy − z = xf1 − f2
is in I, but its initial term yz 5 does not lie in the ideal generated by in< f1 and in< f2 .
Thus {f1 , f2 } is not a Gröbner basis with respect to this term order. One can check
that {f1 , f2 , f3 } is a Gröbner basis for I.
9
Algorithms 2.8 and 2.9 form the computational backbone of the Gröbner basis
approach.
Algorithm 2.8 (Division algorithm). Fix a term order < on C[x1 , . . . , xn ], and a
Gröbner basis G = {g1 , . . . , gr }.
input: A polynomial f ∈ C[x1 , . . . , xn ].
output: The remainder upon division of f by G.
1. Let m be the largest monomial of f which has not yet been processed.
2. If in< gi divides m for some i = 1, . . . , r (say, m′ in< gi = m for some monomial
m′ ), replace f with f − m′ g. Only monomials which are smaller than m are
introduced into f .
3. Add m to the list of processed monomials.
4. If no monomial in f is divisible by any in< gi for i = 1, . . . , r, output f . Otherwise, return to step 1.
Starting with a set of polynomials, one obtains a Gröbner basis for the ideal they
generate by the following algorithm.
Algorithm 2.9 (Buchberger’s algorithm). Fix a term order < on C[x1 , . . . , xn ].
input: A set S := {f1 , . . . , fr } ⊂ C[x1 , . . . xn ] of polynomials which generate the ideal
I ⊂ C[x1 , . . . xn ], with mi := in< fi the initial term of fi (i = 1, . . . , r). Assume that
the coefficient of mi is 1 for each i = 1, . . . , r.
output: A Gröbner basis for I.
1. For each i, j ∈ [r], form the S-polynomial
S(fi , fj ) :=
lcm(mi , mj )
lcm(mi , mj )
fi −
fj
mi
mj
and let fi,j be the reduction of S(fi , fj ) modulo f1 , . . . , fr .
10
2. If fi,j 6= 0, replace S with the set S ∪ {fi,j }, and return to the first step.
3. If each S-polynomial reduces to zero modulo S, output S.
The S-polynomials in Buchberger’s algorithm have the following property.
Proposition 2.10. Let mi = in< fi for i = 1, 2 and polynomials fi ∈ C[x1 , . . . , xn ].
If lcm(m1 , m2 ) = 1, then S(f1 , f2 ) reduces to zero modulo {f1 , f2 }.
Proposition 2.10 will be useful for the polynomial systems we study in Chapter III.
2. Representation theory
We review the basic facts we need from representation theory. See [1, 13] for a more
complete account.
Definition 2.11. An (affine) algebraic group is an (affine) algebraic variety G which
is a group, such that the maps
µ:G×G→G
ι : G −→ G
given by µ(g, h) = gh and ι(g) = g −1 are regular.
Let G be an algebraic group. For any g ∈ G, the map Lg : G → G, Lg (h) := gh
is an automorphism of G (as an algebraic variety). In particular, G is smooth, and
the differential dh Lg : Th G → Tgh G is an isomorphism for all h, g ∈ G. This allows us
to identify the tangent space of G at any point with the tangent space at the identity
g := Te G.
11
For any group G, let [G, G] := {ghg −1 h−1 | g, h ∈ G} denote its commutator
subgroup. The derived series of subgroups is defined inductively by D0 (G) := G and
Di+1 (G) := [Di (G), Di (G)].
Definition 2.12. A group is solvable if Dn (G) = {e} for some n ∈ Z+ .
Definition 2.13. Let G be an algebraic group. A Borel subgroup of G is a maximal
closed connected solvable subgroup. A torus is a closed connected abelian subgroup.
The subgroup B of upper-triangular matrices in G = GLn (C) is a Borel subgroup,
and the diagonal matrices T ⊂ B is a maximal torus (in G or in B; since an abelian
group is solvable, any torus is contained a Borel subgroup). When working with more
general groups G, it is typically the case that there is a natural embedding of G into
GLn (C) such that the upper-triangular matrices in G form a Borel subgroup of G,
and similarly for a maximal torus.
Definition 2.14. A parabolic subgroup of an algebraic group is a closed subgroup
containing some Borel subgroup.
Proposition 2.15. Any two Borel subgroups are conjugate.
Definition 2.16. Let G be an algebraic group. The radical R(G) of G is the intersection of all Borel subgroups of G. We say that G is semisimple if R(G) = {e}.
Proposition 2.17. Let G be an algebraic group and P a closed subgroup. Then P is
parabolic if and only if G/P is a projective variety.
The spaces G/P are called flag varieties. The flag variety G/B can be identified
with the set of all Borel subgroups of G. Since any parabolic subgroup P ⊂ G is
contained in a Borel subgroup B, we have a surjective G-equivariant (regular) map
G/B → G/P . Proposition 2.15 implies that G acts transitively on G/B and hence
also on G/P .
12
Definition 2.18. A rational representation of G is a a vector space V together with
a regular group homomorphism G → GL(V ).
All representations we will consider are rational.
Example 2.19. Let g = Te G be the tangent space at the identity of the algebraic
group G. This is the Lie algebra of G. The (non-associative) algebra structure can
be defined by identifying g with the set of left multiplication-invariant vector fields
on G, but we will not use this.
The Lie algebra g is also a representation of G, called the adjoint representation. For each g ∈ G, the inner automorphism φg (h) := ghg −1 sends e to itself, so
that the differential at e is an invertible linear map de φg : g → g. Thus the assignment g 7→ de φg defines a map G → GL(g), which one can show is a regular group
homomorphism.
Let G be an semisimple affine algebraic group and choose a parabolic subgroup
P ⊂ G, Borel subgroup B ⊂ P , and maximal torus T ⊂ B. Let X(T ) := Hom(T, C∗ )
(where C∗ denotes the multiplicative group of nonzero complex numbers) be the
weight lattice of T . Let N := {g ∈ G | gtg −1 , for all t ∈ T } be the normalizer of T ,
and let W := N/T be the Weyl group.
Proposition 2.20. Let T ⊂ B ⊂ G be as above. Let WP the Weyl group of P . Then
G = BW B and P = BWP B.
Over C, any representation V of G decomposes into a direct sum of irreducible
representations. For a maximal torus T ⊂ G, we call an element ω ∈ X(T ) a weight
of V if there exists a weight vector v ∈ V such that tv = ω(t)v for all t ∈ T . If G is
L
semisimple then V is a direct sum ω∈X(T ) Vω of weight spaces for the action of T .
The Weyl group acts on the set of weights by setting (w ·ω)(t) = ω(ntn−1 ), where
n ∈ N is any coset representative of w ∈ W .
13
The weights of the adjoint representation are called the roots of G. We have
Proposition 2.21. The adjoint representation has the weight space decomposition
L
g = t ⊕ ρ∈Φ gρ , where t is the subalgebra of vectors of weight zero and the set of
nonzero weights Φ ⊂ X(T ) is finite.
The subalgebra t in Proposition 2.21 is in fact the Lie algebra of the maximal
torus T .
Definition 2.22. The set Φ ⊂ X(T ) of non-zero weights of g is called the root system
of G (given T ).
Proposition 2.23. The root system Φ has the following properties:
1. Φ spans X(T )Q := X(T ) ⊗ Q and does not contain 0.
2. For each ρ ∈ Φ there is a reflection sρ leaving Φ stable.
3. For ρ, σ in Φ, sρ (σ) = σ − nρσ ρ for some nρσ ∈ N.
Each reflection can be identified with an element of the Weyl group W , and the set
{sρ | ρ ∈ Φ} generates W .
The vector space X(T )Q admits an inner product invariant under the action of
the Weyl group, which we denote (·, ·). It is called the Killing form.
Proposition 2.24. The Killing form satisfies nρσ = 2(ρ, σ)/(ρ, ρ).
Fixing a Borel subgroup B, we have simple roots S := {ρ1 , . . . , ρr }, where r =
dimX(T )Q .
Proposition 2.25. The set of simple roots has the following properties:
1. S is a basis of X(T )Q .
14
2. W is generated by {sρ | ρ ∈ S}.
For j = 1, . . . , r, let ωj ∈ X(T ) be the unique element such that (ρi , ωj ) = δij
(where δij is the Kronecker delta function). The weights ω1 , . . . , ωr are the fundamental weights.
Fix T ⊂ B ⊂ G, where G is an affine connected semisimple complex algebraic
group, B a Borel subgroup, and T a maximal torus. For any irreducible representation
V , there exists a unique weight ω of V such that BVω ⊂ Vω , called the highest weight
of V . If V ′ is another irreducible representation of highest weight ω then V ′ ∼
= V . We
denote the irreducible representation of highest weight ω by L(ω). Proposition 2.26
gives an identification of the G-orbit of a highest weight vector in L(ωi ) with the flag
variety G/Pi .
Proposition 2.26. Let T ⊂ B ⊂ G be as above. For each non-negative integer linear
combination ω of the fundamental weights, there exists a unique (up to isomorphism)
irreducible representation L(ω) of G.
In particular, for each fundamental weight ωi (1 ≤ i ≤ r) there is a unique
irreducible representation of highest weight ωi . In this case, the stabilizer of a highest
weight vector is a parabolic subgroup Pi ⊂ B.
We will need some facts about the representation theory of the groups SLn+1 (C)
and Sp2n (C). We first consider the special case SL2 (C). For any vector space V
V
and any positive integer m, let S m V denote the mth symmetric power and let m V
denote the mth exterior power of V .
Proposition 2.27. The irreducible representations of SL2 (C) are S d V , where d ∈ Z+
and V = C2 is the defining representation. The following isomorphisms commute with
15
the action of SL2 (C).
a
S 2 (S a V ) ∼
=
2
^
(S a V ) ∼
=
⌊2⌋
M
S 2a−4k V
k=0
⌊ a−1
⌋
2
M
S 2a−4k−2 V
k=0
The summands on the right-hand sides of these isomorphism are irreducible.
Definition 2.28. A partition is a set of boxes (unit squares), arranged in rows and
left-justified, such that as one moves from the top to the bottom the number of boxes
in each row weakly decreases. We denote the partition with k rows and mi boxes in
P
the ith row (starting from the top) by (m1 , . . . , mk ), and define |λ| := ki=1 mi .
Let λ := (λ1 , . . . , λr ) be a partition and λt the transpose (or conjugate) partition
N
to λ. The Schur module [13, 43] Lλ V is a quotient of the tensor product i,j Vi,j ,
(where there is a vector space Vi,j ∼
= V for each box of λ). It is isomorphic to the
image of the homomorphism
φλ :
λ1
^
V ⊗ ··· ⊗
λr
^
t
t
V → S λ1 V ⊗ · · · ⊗ S λr V
defined as the composition of the exterior diagonal
δ:
λ1
^
V ⊗ ··· ⊗
λr
^
V →
O
Vi,j
i,j
followed by the multiplication map
m:
O
i,j
t
t
Vi,j → S λ1 V ⊗ · · · ⊗ S λr V .
Proposition 2.29. Let V = Cn+1 be the defining representation of SLn+1 (C). For
each partition λ = (nan , . . . , 1a1 ) with at most n columns (that is, λ has ai rows of
length i, for each i ∈ [n]), the Schur module Lλ V is isomorphic to the irreducible
16
representation of highest weight ω :=
Pn
i=1
ai ωi .
The irreducible representations of Sp2n (C) have a similar description, as we will
see in Chapter III, Section B.3.
B. Flag varieties
Let G be a semisimple linear algebraic group. Fix a Borel subgroup B ⊂ G and
a maximal torus T ⊂ B. Let R be the set of roots (determined by T ), and S :=
{ρ1 , . . . , ρr } the simple roots (determined by B). The simple roots form an ordered
basis for the Lie algebra t of T . Let {ω1 , . . . , ωr } be the dual basis (the fundamental
weights). The Weyl group W is the normalizer of T modulo T itself.
Let P ⊂ G be the maximal parabolic subgroup associated to the fundamental
weight ω ∈ Ω, let L(ω) be the irreducible representation of highest weight ω, and
let (, ) denote the Killing form on t. For ρ ∈ R, let ρ∨ := 2ρ/(ρ, ρ). For simplicity,
assume that (ω, ρ∨ ) ≤ 2 for all ρ ∈ S (i.e., P is of classical type; see [23]). This
condition implies that L(ω) has T -fixed lines indexed by certain pairs of elements of
W/WP , called admissible (Definition 2.52).
We review these ideas in more detail for the special case of the Lagrangian
Grassmannian.
1. The Lagrangian Grassmannian
In its natural projective embedding, the Lagrangian Grassmannian LG(n) is defined
by quadratic relations which give a straightening law on a certain ordered set [9].
These relations are obtained by expressing LG(n) as a linear section of Gr(n, 2n).
While this is well-known, the author knows of no explicit derivation of these relations
which do not require the representation theory of semisimple algebraic groups. We
17
provide a derivation which does not rely upon representation theory (although we
adopt the same notation and terminology) which will be useful when we consider the
Drinfel’d Lagrangian Grassmannian, for which representation theory has yet to be
successfully applied.
Set [n] := {1, 2, . . . , n}, ı̄ := −i, and hni := {n̄, . . . , 1̄, 1, . . . n}. If S is any set,
¡S ¢
let k be the collection of subsets α = {α1 , . . . , αk } of cardinality k.
V
The Grassmannian Gr(n, 2n) is the subvariety of P( n C2n ) defined by the Plücker
V
relations. The projective space P( n C2n ) has Plücker coordinates indexed by the dis¡[2n]¢
¡ ¢
.
.
Let
∧
and
∨
denote
the
meet
and
join
in
tributive lattice [2n]
n
n
Proposition 2.30 ([12, 19]). For α, β ∈
pα pβ − pα∧β pα∨β +
The defining ideal of Gr(n, 2n) ⊂ P(
¡[2n]¢
n
X
there is a Plücker relation
cγ,δ
α,β pγ pδ = 0 .
γ<α∧β<α∨β<δ
Vn
C2n ) is generated by the Plücker relations.
Fix an ordered basis {en̄ , . . . , e1̄ , e1 , . . . , en } of the vector space C2n , and let
Pn
Ω :=
i=1 eı̄ ∧ ei be a non-degenerate alternating bilinear form. The Lagrangian
Grassmannian LG(n) is the set of maximal isotropic subspaces (relative to Ω) of C2n .
Let {hi := Eii − Eı̄ı̄ | i ∈ [n]} be the usual basis for the Lie algebra t of T [13]
and {h∗i | i ∈ [n]} ⊂ t∗ the dual basis. Observe that h∗ı̄ = −h∗i . The weights of
any representation of Sp2n (C) are Z-linear combinations of the fundamental weights
ωi := h∗n−i+1 + · · · + h∗n .
V
V
The weights of both representations L(ωn ) ⊂ n C2n and n C2n are of the form
¡ ¢
P
. If αj = ᾱj ′ for some j, j ′ ∈ [n], then h∗αj = −h∗αj′ ,
ω = ni=1 h∗αi for some α ∈ hni
n
and thus the support of ω does not contain h∗αj . Hence the set of all such ω are
¡ ¢
such that k = 1, . . . , n and n−k is even.
indexed by elements of hni
k
V
Let V be a vector space. For simple alternating tensors v := v1 ∧ · · · ∧ vl ∈ l V
18
V
and ϕ := ϕ1 ∧ · · · ∧ ϕk ∈ k V ∗ , there is a contraction, defined by setting


 P [l] ±v1 ∧ · · · ∧ ϕ1 (vi ) ∧ · · · ∧ ϕk (vi ) ∧ · · · ∧ vl , k ≤ l
1
k
I∈( k )
ϕyv :=


0,
k>l
V
V
V
and extending bilinearly to a map k V ∗ ⊗ l V → l−k V . In particular, for a fixed
V
V
V
element Φ ∈ k V ∗ , we obtain a linear map Φy• : l V → l−k V .
The Lagrangian Grassmannian embeds in PL(ωn ), where L(ωn ) is the irreducible
Sp2n (C)-representation of highest weight ωn = h∗1 + · · · + h∗n . By Proposition 2.31,
V
this representation is isomorphic to the kernel of the contraction Ωy• : n C2n →
Vn−2 2n
C . We thus have a commutative diagram of injective maps:
LG(n) −−−→ Gr(n, 2n)




y
y
Vn 2n
PL(ωn ) −−−→ P( C ).
The next proposition implies that LG(n) = Gr(n, 2n) ∩ PL(ωn ).
Proposition 2.31. The dual of the contraction map
Ωy• :
n
^
2n
C
→
n−2
^
C2n
is the multiplication map
Ω∧•:
n−2
^
2n ∗
C
→
n
^
∗
C2n .
Furthermore, the irreducible representation L(ωn ) is defined by the vanishing of the
linear forms
Ln := hΩ ∧ e∗α1 ∧ · · · ∧ e∗αn−2 | α ∈
¡ hni ¢
n−2
i.
These linear forms cut out LG(n) scheme-theoretically in Gr(n, 2n). Dually, L(ωn ) =
ker(Ωy•).
19
Proof. Statement (1) is straightforward, and (2) can be found in [43, Ch. 3, Exercise
1; Ch. 6, Exercise 24].
Since the linear forms in Ln are supported on variables indexed by α ∈
¡hni¢
n
such that {ı̄, i} ∈ α for some i ∈ [n], the set of complementary variables is linearly
¡ ¢
independent. These are indexed by the set Pn of admissible elements of hni
n
Pn :=
©
α∈
¡hni¢
n
ª
| i ∈ α ⇔ ı̄ 6∈ α ,
and have a nice description in terms of partitions (Definition 2.28).
Consider the lattice Z2 with coordinates (a, b) corresponding to the point a units
below the origin and b units to the right of the origin. Given an increasing sequence
¡ ¢
, let [α] be the lattice path beginning at (n, 0), ending at (0, n), and whose
α ∈ hni
n
ith step is vertical if i ∈ α and horizontal if i 6∈ α. We can associate a partition to
α by taking the boxes lying in the region bounded by the coordinate axes and [α].
¡ ¢
is associated to the partition shown in
For instance, the sequence α = 4̄2̄23 ∈ h4i
4
Figure 1.
Proposition 2.32. The bijection between increasing sequences and partitions induces
a bijection between sequences α which do not contain both i and ı̄ for any i ∈ [n], and
partitions which lie inside the n × n square (nn ) and are symmetric with respect to
reflection about the diagonal {(a, a) | a ∈ Z} ⊂ Z2 .
Proof. Pn consists of the α ∈
¡hni¢
n
which are fixed upon negating each element of α
and taking the complement in hni. On the other hand, the composition of these two
operations corresponds to reflecting the associated diagram about the diagonal.
20
4
(4, 0)
3
1̄
3̄
2
1
2̄
4̄
(0, 4)
Fig. 1. The partition (3, 3, 1) associated to 4̄2̄23.
Remark 2.33. We will use an element of
¡hni¢
n
and its associated partition inter-
changeably. We denote by αt the transpose partition obtained by reflecting α about
the diagonal in Z2 . As a sequence, αt is the complement of {ᾱ1 , . . . , ᾱn } ⊂ hni.
Definition 2.34. The Lagrangian involution is the map τ : pα 7→ σα pαt , where
αt is the sequence obtained by negating each element of the complement of α, and
c
c
σα := sgn(α+
, α+ )sgn(α− , α−
) ∈ {±1}.
We denote by α+ (respectively, α− ) the subset of positive (respectively, negative)
elements of α. For example, if α = 4̄1̄23, then α+ = 23 and α− = 4̄1̄.
The Grassmannian Gr(n, 2n) has a natural geometric involution •⊥ : Gr(n, 2n) →
Gr(n, 2n) sending an n-plane U to its orthogonal complement U ⊥ := {u ∈ C2n |
Ω(u, u′ ) = 0, for all u′ ∈ U } with respect to Ω. The next proposition relates •⊥ to
the Lagrangian involution.
21
Proposition 2.35. The map •⊥ : Gr(n, 2n) → Gr(n, 2n) expressed in Plücker coordinates coincides with the Lagrangian involution:
h
¯
¯
h
¡ ¢i
¡hni¢i
¯
¯
.
7→ σα pαt ¯ α ∈ hni
pα ¯ α ∈ n
n
(2.36)
In particular, the relation pα − σα pαt holds on LG(n).
Proof. The set of n-planes in C2n which do not meet the span of the first n standard
basis vectors is open and dense in Gr(n, 2n). Any such n-plane is the row space of an
n × 2n matrix
Y
:= (I | X)
where I is the n × n identity matrix and X is some n × n matrix. We work in
¡ ¢
, we denote the αth
the affine coordinates given by the entries in X. For α ∈ hni
n
minor of Y by pα (Y ). For a set of indices α = {α1 , . . . , αk } ⊂ [n], let αc := [n] \ α
be the complement, α′ := {n−αk +1, . . . , n−α1 +1}, and ᾱ := {ᾱ1 , . . . , ᾱk }. Via the
correspondence between partitions and sequences (Proposition 2.32), αt = ᾱc .
We claim that •⊥ reflects X along the antidiagonal. To see this, we simply
observe how the rows of Y pair under Ω. For u, v ∈ Cn , we denote the concatenation
by (u, v) ∈ C2n . Let ri := (ei , vi ) ∈ C2n be the ith row of Y . For k ∈ hni, we let
rik ∈ C be the k th component of ri . Then, for i, j ∈ hni,
Ω(ri , rj ) = (ei , vi ) · (−vj , ej )t
= ri,n−j+1 − rj,i .
It follows that the effect of τ on the minor Xρ,γ of X given by row indices ρ and
column indices γ is
τ (X)ρ,γ = Xγ ′ ,ρ′ .
22
Let α = ǭ ∪ φ ∈
¡hni¢
n
, where ǫ and φ are subsets of [n]. Combining this description of
τ with the identity
pα (Y ) = sgn(ǫc , ǫ)X(ǫc )′ ,φ ,
we have
pα (τ (Y )) = sgn(ǫc , ǫ)τ (X)(ǫc )′ ,φ
= sgn(ǫc , ǫ)Xφ′ ,ǫc
= sgn(ǫc , ǫ)sgn(φ, φc )p(φ̄c ,ǫc ) (Y )
= σα pαt (Y ) .
It follows that the relation pα − σα pαt = 0 holds on a dense open subset and hence
identically on all of LG(n).
It follows from Proposition 2.35 that the system of linear equations
L′n := hpα − σα pαt | α ∈
defines LG(n) ⊂ Gr(n, 2n) set-theoretically.
¡hni¢
n
i
Since LG(n) lies in no hyperplane
of PL(ωn ), L′n is a linear subspace of the span Ln of the defining equations of
V
L(ωn ) ⊂ n C2n . The generators of L′n above suggests that homogeneous coordinates for the Lagrangian Grassmannian should be indexed by some sort of quotient
(which we will call Dn ) of the poset Pn . The correct notion of quotient is that of a
doset (Definition 2.43). An important set of representatives for Dn in Pn is given in
Proposition 2.38.
By Proposition 2.32, we may identify elements of
¡hni¢
n
with partitions lying in
the n × n square (nn ), and Pn with the set of symmetric partitions. Define a map
¡ ¢
→ Pn × Pn by πn (α) := (α ∧ αt , α ∨ αt ). Let Dn be the image of πn . It is
πn : hni
n
called the set of admissible pairs. It is a subset of OPn := {(α, β) ∈ Pn × Pn | α ≤ β}.
23
The image of Pn ⊂
¡hni¢
n
under πn is the diagonal ∆Pn ⊂ Pn × Pn .
To show that Dn indexes coordinates on LG(n), we will work with a convenient
set of representatives of the fibers of πn . The fiber of (α, β) ∈ Dn can be described
as follows. The lattice paths [α] and [β] must meet at the diagonal. Since α and
β are symmetric, they are determined by the segments of their associated paths to
the right and above the diagonal. Let Π(α, β) be the set of boxes bounded by these
segments. Taking n = 4 for example, Π(4̄2̄13, 3̄1̄24) consists of the two shaded boxes
above the diagonal in Figure 2. The lattice path [4̄2̄13] is above and to the left of the
path [3̄1̄24].
4̄2̄13
3̄1̄24
Fig. 2. The paths associated to 4̄2̄13 and 3̄1̄24 in P4 .
A subset S ⊂ (nn ) of boxes is disconnected if S = S1 ⊔ S2 and no box of S1 shares
an edge with a box of S2 . A subset S is connected if it is not disconnected. Let
F
Π(α, β) = ki=1 Si be the decomposition of Π(α, β) into its connected components.
Any element γ of the fiber πn−1 (α, β) is obtained by choosing a subset I ⊂ [k]
24
and setting
γ = α∪
³[
i∈I
´ ³[ ´
Sit ∪
Si .
i6∈I
The elements of πn−1 (4̄2̄13, 3̄1̄24) are shown in Figure 3.
For any diagram α ⊆ (nn ), let α+ ⊆ α be the subset of α on or above the main
diagonal, and α− ⊆ α the subset of α on or below the main diagonal. Equivalently,
α is a strictly increasing subsequence of (n̄, . . . , 1̄, 1, . . . , n) of length n, α+ is the
subset of positive elements of α, and α− is the set of negative elements. Similarly, let
Π+ (α, β) ⊂ Π(α, β) be the set of boxes above the diagonal.
4̄2̄24 and 3̄1̄13
4̄1̄14 and 3̄2̄23
Fig. 3. Elements of πn−1 (4̄2̄13, 3̄1̄24).
The following definition is essential.
t
Definition 2.37. A partition α Northeast if α−
⊆ α+ and Southwest if its transpose
is Northeast.
25
For example, 4̄2̄24 ∈ P4 is Northeast while 4̄1̄14 ∈ P4 is neither Northeast nor
Southwest.
We summarize these ideas in the following proposition.
Proposition 2.38. Let (α, β) ∈ Dn be an element of the image of πn . Then πn−1 (α, β)
is in bijection with the set 2Π+ (α,β) of subsets of connected components of Π+ (α, β).
There exists a unique Northeast element of πn−1 (α, β); namely, the element corresponding to all the connected components. Similarly, there is a unique Southwest
element corresponding to the empty set of components.
Example 2.39. The Lagrangian Grassmannian LG(4) ⊂ Gr(4, 8) is defined by the
linear ideal L4 . From the explicit generators given in Proposition 2.31, it is evident
V
∗
that L4 ⊂ 4 C8 is spanned by weight vectors. For example, the generators of
V
∗
L4 ∩ ( 4 C8 )0 are vectors of weight zero:
Ω ∧ p1̄1 = p4̄1̄14 + p3̄1̄13 + p2̄1̄12
Ω ∧ p2̄2 = p4̄2̄24 + p3̄2̄23 + p2̄1̄12
(2.40)
Ω ∧ p3̄3 = p4̄3̄34 + p3̄2̄23 + p3̄1̄13
Ω ∧ p4̄4 = p4̄3̄34 + p4̄2̄24 + p4̄1̄14
The following linear forms lie in the span of the right-hand side of (2.40):
p2̄1̄12 + p4̄3̄34
p3̄1̄13 + p4̄2̄24
p3̄2̄23 + p4̄1̄14
(2.41)
p4̄1̄14 + p4̄2̄24 + p4̄3̄34
Three of the linear forms in (2.41) are supported on a pair {pα , pαt }, and the remaining
linear form expresses the Plücker coordinate p4̄1̄14 as a linear combination of coordinates indexed by Northeast partitions (in general, this follows from Lemma 3.10).
Since each pair {pα , pαt } is incomparable, there is a Plücker relation which, after
26
reduction by the linear forms (2.40), takes the form
±p2α − pβ pγ + lower order terms
where β := α ∧ αt and γ := α ∨ αt are respectively the meet and join of α and αt .
Defining p(β,γ) := pα = σα pαt we can regard such an equation as giving a rule for
rewriting p2(β,γ) as a linear combination of monomials supported on a chain. These
facts are proven in general in Chapter III.
C. Spaces of algebraic maps
The flag variety G/P embeds in PL(ω) as the orbit of a highest weight vector. Define
the degree of a rational map f : P1 → G/P to be its degree as a rational map into
PL(ω).
Let Md (G/P ) be the space of algebraic maps of degree d from P1 into G/P .
If P is of classical type then the set D of admissible pairs indexes the homogeneous
coordinates on PL(ω) (see Definition 2.52, and for more background [35, 23]). Therefore, any map f ∈ Md (G/P ) can be expressed as f : [s, t] 7→ [pw (s, t) | w ∈ D],
where D is the set of admissible pairs and pw (s, t) is a homogeneous form of degree
d. This leads to an embedding of Md (G/P ) in P(S d C2 ⊗ L(ω)), where S d C2 is the
space of homogeneous polynomials of degree d in two variables. Coordinate functions
on S d C2 ⊗ L(ω) are indexed by {w(a) | w ∈ D, a = 0, . . . , d}, a disjoint union of
d+1 copies of D. The closure of Md (G/P ) ⊂ P(S d C2 ⊗ L(ω)) is called the Drinfel’d
compactification and denoted Qd (G/P ). This definition is due to V. Drinfel’d, dating
from the mid-1980s. Drinfel’d never published this definition himself; to the author’s
knowledge its first appearance in print was in [36]; see also [22].
Let G = SLn (C) and P be the maximal parabolic subgroup stabilizing a k-
27
dimensional subspace, so that G/P = Gr(k, n). In this case we denote the Drinfel’d Grassmannian Qd (G/P ) by Qd (k, n). In [41] it is shown that the homogeneous
coordinate ring of Qd (k, n) is an algebra with straightening law on a distributive
¡ ¢
d
, hence normal, Cohen-Macaulay, and Koszul, and its ideal Ik,n−k
has a
lattice [n]
k d
d
quadratic Gröbner basis consisting of the straightening relations. The ideal Ik,n−k
¡ ¢
] whose variables are indexed by elements
is a subset of the polynomial ring C[ [n]
k d
¡[n]¢
¡
¢
¡[n]¢
of k d := {α(a) | α ∈ [n]
,
0
≤
a
≤
d}.
The
partial
order
on
is defined by
k
k d
α(a) ≤ β (b) if and only if a ≤ b and αi ≤ βb−a+i for i = 1, . . . , k−b+a.
Note that taking d = 0 above, one recovers the classical Bruhat order on
¡[n]¢
k
. For
an arbitrary semisimple algebraic group, this is an ordering on the set of maximal coset
representatives of the quotient of the Weyl group by the Weyl group of a parabolic
subgroup.
Suppose that d = ℓk+q for positive integers ℓ and q, and let X = (xij )1≤i,j≤n be
(k )
(1)
(0)
a matrix of polynomials with xij = xij i tki + · · · + xij t + xij , ki = ℓ+1 for i ≤ q,
¡ ¢
d
and ki = ℓ for i > q. The ideal Ik,n−k
is the kernel of the map ϕ : C[ [n]
] → C[X]
k d
(a)
sending the variable pα indexed by α(a) to the coefficient of ta in the maximal minor
of X whose columns are indexed by α.
The main results of [41] follow from the next theorem. Given any distributive
lattice, we denote by ∧ and ∨, respectively, the meet and join. The symbol ∧ will
also be used for exterior products of vectors, but the meaning should always be clear
from the context.
¡ ¢
Theorem 2.42 ([41]). Let γ, ǫ be a pair of incomparable variables in the poset [n]
.
k d
¡[n]¢
There is a quadratic polynomial S(γ, ǫ) in the kernel of ϕ : C[ k d ] → C[X] whose
first two monomials are
pγ pǫ − pγ∧ǫ pγ∨ǫ .
28
Moreover, if λpβ pα is any non-initial monomial in S(γ, ǫ), then γ, ǫ lies in the interval
¡ ¢
| β ≤ γ ≤ α}.
[β, α] = {γ ∈ [n]
k d
The quadratic polynomials S(γ, ǫ) in fact form a Gröbner basis for the ideal they
generate. It is shown in [41] that there exists a toric (sagbi) deformation taking
S(γ, ǫ) to its initial form pγ pǫ − pγ∧ǫ pγ∨ǫ .
Our main goal is to extend this result to the Lagrangian Drinfel’d Grassmannian
LQd (n) := Qd (LG(n)) parametrizing rational curves of degree d in the Lagrangian
Grassmannian.
The role of flag varieties in classical enumerative geometry was described in
Chapter I. Drinfel’d flag varieties are similarly useful in quantum cohomology, which
deals with the enumerative geometry of curves in a flag variety. We illustrate this for
the case of the Drinfel’d Grassmannian [39, 41].
Let Xα ⊂ Gr(k, n) be a Schubert variety and s ∈ P1 a point. An element of the
Drinfel’d Grassmannian E(t) ∈ Qd (k, n), is a parametrized curve t 7→ E(t) ∈ Gr(k, n)
of degree d. Given a non-negative integer a ≤ d, the set of all E(t) ∈ Qd (k, n)
such that E(t) meets Xα to order a at the point s is a Schubert variety Xα(a) (s).
When s = 0 ∈ P1 , this coincides with the definition given in Section D, below.
The realization of the coordinate ring of a Drinfel’d flag variety (and its Schubert
varieties) allows one to compute the degree of a Schubert variety, thus recovering
certain intersection numbers in quantum cohomology.
D. Algebras with straightening laws
The following definitions are fundamental.
Let P be a poset, ∆P the diagonal in P ×P, and OP := {(α, β) ∈ P ×P | α ≤ β}
the subset of P × P defining the order relation on P.
29
Definition 2.43. A doset on P is a set D such that ∆P ⊂ D ⊂ OP , and if α ≤ β ≤ γ,
then (α, γ) ∈ D if and only if (α, β) ∈ D and (β, γ) ∈ D. The ordering on D is given
by (α, β) ≤ (γ, δ) if and only if β ≤ γ in P. We refer to P as the underlying poset.
Remark 2.44. A doset is not usually a poset: the ordering is non-reflexive except
in the trivial case when D = ∆P . In any case, note that we may regard P ∼
= ∆P as
a subset of D.
The Hasse diagram of a doset D on P is obtained from the Hasse diagram of
P ⊂ D by drawing a double line for each cover α ⋖ β such that (α, β) is in D. The
defining property of a doset implies that we can recover all the information in the
doset from its Hasse diagram. See Figure 6 for an example.
Loosely, an algebra with straightening law is an algebra generated by elements
{pα | α ∈ D} indexed by a (finite) doset with a basis consisting of standard monomials supported on a chain. That is, a monomial is standard if it has the form
p(α1 ,β1 ) · · · p(αk ,βk ) , where α1 ≤ β1 ≤ α2 ≤ β2 ≤ · · · ≤ αk ≤ βk . Furthermore,
monomials which are not standard are subject to so-called straightening relations, as
described in the following definition.
Definition 2.45. Let D be a doset. A graded C-algebra A =
L
q≥0
Aq is an algebra
with straightening law on D if there is an injection D ∋ (α, β) 7→ p(α,β) ∈ A such that:
1. {p(α,β) | (α, β) ∈ D} generates A.
2. The set of standard monomials are a C-basis of A.
3. For any monomial m = p(α1 ,β1 ) · · · p(αk ,βk ) , (αi , βi ) ∈ D for i = 1, . . . , k, if
m=
N
X
j=1
cj p(αj1 ,βj1 ) · · · p(αjk ,βjk ) ,
30
is the unique expression of m as a linear combination of distinct standard monomials, then the sequence (αj1 ≤ βj1 ≤ · · · ≤ αjk ≤ βjk ) is lexicographically
smaller than (α1 ≤ β1 ≤ · · · ≤ αk ≤ βk ). That is, if ℓ ∈ [2k] is minimal such
that αjℓ 6= αℓ , then αjℓ < αℓ .
4. If α1 ≤ α2 ≤ α3 ≤ α4 are such that for some permutation σ ∈ S4 we have
(ασ(1) , ασ(2) ) ∈ D and (ασ(3) , ασ(4) ) ∈ D, then
p(ασ(1) ,ασ(2) ) p(ασ(3) ,ασ(4) ) = ±p(α1 ,α2 )(α3 ,α4 ) +
N
X
ri m i
(2.46)
i=1
where the mi ∈ A2 are standard monomials distinct from p(α1 ,α2 ) p(α3 ,α4 ) .
The ideal of straightening relations is generated by homogeneous quadratic forms
in the pα (α ∈ D), so we may consider the projective variety X := Proj A they define.
For each α ∈ P, we have the Schubert variety
Xα := {x ∈ X | p(β,γ) (x) = 0 for γ 6≤ α}
and the dual Schubert variety
X α := {x ∈ X | p(β,γ) (x) = 0 for β 6≥ α}.
Let us recall the geometry of these varieties for the Grassmannian of k-planes in Cn ,
¡ ¢
.
whose coordinate ring is an algebra with straightening law on the poset [n]
k
For each i ∈ [n], set Fi := he1 , . . . , ei i and Fi′ := hen , . . . , en−i+1 i, where h · · · i
denotes linear span and {e1 , . . . , en } is the standard basis of Cn . We call F• := {F1 ⊂
· · · ⊂ Fn } the standard coordinate flag, and F•′ := {F1′ ⊂ · · · ⊂ Fn′ } the opposite flag.
¡ ¢
, define Xα := {E ∈ Gr(k, n) | dim(E ∩
For α = {1 ≤ α1 < · · · < αk ≤ n} ∈ [n]
k
′
) ≤ i, i = 1, . . . , k} and X α := {E ∈ Gr(k, n) | dim(E ∩Fα′ i ) ≤ i, i = 1, . . . , k}.
Fn−α
i +1
We represent any k-plane E ∈ Gr(k, n) as the row space of a k × n matrix.
31
Furthermore, any such k-plane E is the row space of a unique reduced row echelon
matrix. The Schubert variety Xα consists of precisely the k-planes E such that the
pivot in row i is weakly to the right of column n−αi +1. Since the Plücker coordinate
pα (E) is just the maximal minor of this matrix, we see that E ∈ Xα if and only if
pβ (E) = 0 for all β 6≤ α; hence our definition of Xα for the Grassmannian agrees with
the general definition above.
Schubert varieties are useful and natural tools for studying algebras with straightening law, as we will see in Chapter III.
1. Hilbert series of an algebra with straightening law
We compute the Hilbert series of an algebra with straightening law A on a doset, and
thus obtain formulas for the dimension and degree of Proj A. Let P be a poset and D
a doset on P. Assume that all maximal chains in D (respectively, P) have the same
length d (respectively, p).
We first compute the Hilbert series of A with respect to a suitably chosen fine
grading of A. Namely, A is graded by the elements of a semigroup, defined as follows.
Monomials in C[D] are determined by their exponent vectors. We can therefore
identify the set of such monomials with the semigroup ND . Define the weight map
w : ND → QP by setting w(α, β) :=
ǫα +ǫβ
,
2
where, ǫα ∈ QP (α ∈ P) has α-coordinate
equal to 1 and all other coordinates equal to 0. This gives a (fine) grading of A by
the semigroup im(w). Let Ch(D) be the set of all chains in D. Since the standard
monomials (those supported on a chain) form a C-basis for A, the Hilbert series with
respect to this fine grading is
HA (r) =
X
X
c∈Ch(D) a∈im(w)
supp(a)=c
ra
32
where r := (rα | α ∈ P), a = (aα | α ∈ P), and ra =
Q
α∈P
rαaα . Note that elements of
im(w) correspond to certain monomials with rational exponents (supported on P).
√
For example, (α, β) ∈ D corresponds to rα rβ . Setting all rα = r, we obtain the
usual (coarse) Hilbert series, defined with respect to the usual Z-grading on A by
degree.
Example 2.47. Consider the doset D := {α, (α, β), β} on the two element poset
{α < β}. The elements of Ch(D) are shown in Figure 4.
Ch(D) = {∅, {α}, {β}, {(α, β)}, {α, (α, β)}, {(α, β), β}, {α, β}, {α, (α, β), β}} .
∅
{α}
{α, (α, β)} {(α, β), β}
{β}
{(α, β)}
{α, β} {α, (α, β), β}
Fig. 4. The set Ch(D) of chains in D.
33
We have
3/2 1/2
rα rβ
rβ
rα
√
+
+ rα rβ +
HA (r) = 1 +
1 − r α 1 − rβ
1 − rα
1/2 3/2
3/2 3/2
rα rβ
rα rβ
rα rβ
+
+
+
.
(1 − rβ ) (1 − rα )(1 − rβ ) (1 − rα )(1 − rβ )
Setting r = rα = rβ , we obtain the Hilbert series with respect to the usual Z-grading
of C[D].
2r
2r2
r3 + r2
+r+
+
1−r
1 − r (1 − r)2
r+1
=
(1 − r)2
∞
X
= 1+
(2i+1)ri .
hA (r) = 1 +
i=1
We see that hA (i) = 2i+1, dim(Proj A) = 1, and deg(Proj A) = 2.
Remark 2.48. The Lagrangian Grassmannian LG(2) is an algebra with straightening
law on the five element doset obtained by adding two elements 0̂ < α and 1̂ > β to
the doset D in Example 2.47. A similar computation shows that the degree of LG(2)
is also 2. Thus we confirm our stated answer to Question 1.5. Theorem 3.14 allows us
to carry out such degree computations for the Drinfel’d Lagrangian Grassmannian;
this gives a new derivation of numbers computed in quantum cohomology.
Fix a chain {α1 , . . . , αu , (β11 , β12 ), . . . , (βv1 , βv2 )} ⊂ D (not necessarily written in
order), let ti be the formal variable corresponding to αi (i = 1, · · · , u), and let sjk
correspond to βjk (j = 1, . . . , v, k = 1, 2). The variables r and s are not necessarily
disjoint; in the example above, the chain {α, (α, β)} has r1 = s11 . We have
X
a∈im(w)
supp(a)=c
ra =
u
Y
i=1
v
Y
ri
√
sj1 sj2
·
1 − ri j=1
34
Recall that we may identify P with the diagonal ∆P ⊂ D ⊂ P × P. Letting cvu denote
the number of chains consisting of u elements of P and v elements of D \ P, we have
HSA (r) =
p+1 d−p
X
X
cvu
u=0 v=0
p+1 d−p
=
XX
ru+v
(1 − r)u
∞
X
cvu ru+v (
u=0 v=0
d−p
=
X
cv0 rv
+
v=0
rk )u
k=0
p+1 d−p
∞
XXX
ℓ=0 u=1 v=0
cvu
µ
¶
u+ℓ−1 u+v+ℓ
r
.
u−1
When w > d−p, the coefficient of rw agrees with the Hilbert polynomial:
HPA (w) =
p+1 d−p
X
X
u=1 v=0
cvu
µ
¶
w−v−1
.
u−1
In particular, the dimension of Proj A is p, since this is the largest possible value of
¢
¡
. The leading monomial of HPA (w) is
u − 1 = degw w−v−1
u−1
d−p
X
cvp+1
v=0
µ
¶
w−v−1
.
p−1
By our assumption that the maximal chains in P (respectively, D) have the same
¡ ¢ 0
cp+1 , so that the leading coefficient of HPA (w) is
length, we have cvp+1 = d−p
v
¶
d−p µ
2d−p c0p+1
c0p+1 X d−p
=
,
v
(p−1)! v=0
(p−1)!
from which we deduce the degree of ProjA.
Theorem 2.49. deg(Proj A) = 2d−p c0p .
Example 2.50. Let A be an algebra with straightening law on the doset D shown
35
in Figure 5. We have rk P = 2, rk D = 3, and
Ch(D) =
n
∅, {α}, {β}, {γ}, {δ}, {(α, γ)}, {(β, δ)},
{α, γ}, {α, β}, {α, δ}, {β, δ}, {γ, δ}, {α, (α, γ)},
{α, (β, δ)}, {(α, γ), γ}, {(α, γ), δ}, {β, (β, δ)},
{(β, δ), δ}, {α, β, δ}, {α, γ, δ}, {α, (α, γ), γ},
{α, (α, γ), δ}, {α, β, (β, δ)}, {α, (β, δ), δ},
{β, (β, δ), δ}, {α, (α, γ), γ, δ}, {α, β, (β, δ), δ}
o
and the values of cvu are given by the following matrix, whose entry in row i and
column j is ci−1
j−1 :


 1 4 5 2 


2 6 5 2
The Hilbert polynomial is therefore:
¶
¶
µ
¶
µ
µ
w−1
w−1
w−1
+2
+5
HPA (w) = 4
2
1
0
¶
¶
µ
¶
µ
µ
w−2
w−2
w−2
+2
+5
+ 6
2
1
0
= 2w2 + 2w + 3 = 4
w2
+ 2w + 3.
2!
In particular, deg(Proj A) = 4.
Theorem 3.14 will allow us to compute intersection numbers in quantum cohomology using Theorem 2.49, as illustrated in Example 2.50. The appropriate doset is
described in Subsection 2. We will show that the Drinfel’d Lagrangian Grassmannian
is a algebra with straightening law on this doset.
36
δ
γ
β
α
Fig. 5. The doset of Example 2.50
2. The doset of admissible pairs
We define the doset of admissible pairs on the poset Pd,n . Let us first consider an
example.
Example 2.51. Consider the poset
n
o
¡ ¢
P2,4 := α(a) ∈ h4i
|
i
∈
α
⇐⇒
ı̄
∈
6
α
4 2
of admissible elements of
¡h4i¢
4
2
. The set D2,4 of elements (α, β)(a) ∈ OP2,4 such that
α and β have the same number of negative elements is a doset on P2,4 . The Hasse
diagram for D2,4 is shown in Figure 6.
(a)
To each (α, β)(a) ∈ P2,4 , we have the Plücker coordinate p(α,β) := sa tb ⊗ p(α,β) in
S d C2 ⊗ L(ωn ), where S d C2 is the space of homogeneous polynomials of degree d in
two variables.
37
4̄123(0)
4̄3̄12(0)
4̄3̄1̄2(0)
4̄3̄2̄1(0)
4̄3̄2̄1̄
4̄1̄23(0)
4̄2̄13(0)
4̄2̄1̄3(0)
(0)
1234(0)
3̄124(0)
3̄1̄24(0)
3̄2̄14(0)
2̄134(0)
2̄1̄34(0)
3̄2̄1̄4(0)
1̄234(0)
4̄3̄2̄1̄(1)
4̄3̄12(1)
4̄3̄1̄2(1)
4̄3̄2̄1(1)
4̄123(1)
4̄1̄23(1)
4̄2̄13(1)
4̄2̄1̄3(1)
3̄124(1)
3̄1̄24(1)
3̄2̄14(1)
1234(1)
1̄234(1)
2̄134(1)
2̄1̄34(1)
3̄2̄1̄4(1)
4̄3̄12(2)
4̄3̄1̄2(2)
4̄3̄2̄1(2)
4̄3̄2̄1̄(2)
4̄123(2)
4̄1̄23(2)
4̄2̄13(2)
4̄2̄1̄3(2)
3̄124(2)
3̄1̄24(2)
3̄2̄14(2)
1234(2)
1̄234(2)
2̄134(2)
2̄1̄34(2)
3̄2̄1̄4(2)
Fig. 6. The doset D2,4 .
Let
¡hni¢
n
d
∼
=
¡[2n]¢
n
d
be the poset associated to the Drinfel’d Grassmannian Qd (n, 2n)
from the introduction, and let Pd,n be the subposet consisting of the elements α(a)
such that αt = α. There are three types of covers in Pd,n .
1. α(a) < β (a) , where α and β have the same number of negative elements. For
example, 4̄2̄13(a) < 4̄1̄23(a) ∈ Pd,4 for any non-negative integers a ≤ d.
2. α(a) < β (a) , where the number of negative elements in β is one less than the
number of negative elements of α. For example, 4̄1̄23(a) < 4̄123(a) ∈ Pd,4 for
any non-negative integers a ≤ d.
3. α(a) < β (a+1) , where the number of negative elements of β is one more than the
number of negative elements of α, n̄ ∈ β, and n ∈ α. For example, 3̄2̄14(a) <
4̄3̄2̄1(a+1) for any non-negative integers a ≤ d.
38
The first two types are exactly those appearing in the classical Bruhat order on
(a)
P0,n . It follows that Pd,n is a union of levels Pd,n , each isomorphic to the Bruhat
order, with order relations between levels imposed by covers of the type (3) above.
We define the doset Dd,n of admissible pairs in Pd,n .
Definition 2.52. A pair (α(a) < β (a) ) is admissible if there exists a saturated chain
α = α0 < α1 < · · · < αs = β, where each αi < αi+1 is a cover of type (1).
We denote the set of admissible pairs by Dd,n . Observe that (α(a) < β (b) ) is never
admissible if a < b.
Proposition 2.53. The set Dd,n ⊂ Pd,n × Pd,n is a doset on Pd,n . The poset Pd,n is
a distributive lattice.
Proof. To show that Dd,n is a doset, one may simply apply the proof in the case d = 0
(see [9]) to the subposet consisting of all the elements of the form (α, β)(a) , for each
a = 0, . . . , d to show that Dd,n is a doset on Pd,n .
To prove that Pd,n is a distributive lattice, we realize the meet and join as the
intersection and union of certain sets (ordered by inclusion) generalizing the usual
notion of a partition. Consider the union of of n shifted n × n squares in Z2 .
Sd,n :=
d
[
a=0
{(i−a, j−a) | 0 ≤ i, j ≤ n}
To α(a) ∈ Pd,n , we associate the subset of Sd,n defined by shifting the (open)
squares in α by (−a, −a), and adding the boxes obtained by translating a box of α by
a vector (v1 , v2 ) with v1 , v2 ≤ 0 and the points (−i, −i) for i = 0, . . . , a. See Figure 7
for an example.
39
Fig. 7. The subset of S2,4 associated to 4̄2̄13(1) ∈ P2,4 .
It is straightforward to check that the (symmetric) subsets obtained in this way
form a distributive lattice (ordered by containment) isomorphic to Pd,n .
40
CHAPTER III
RESULTS
A. Algebras with straightening law
Fixing a projective variety X ⊂ Pn , there are many homogeneous ideals I ⊂ C[x0 , . . . , xn ]
such that V(I) = X. However, fixing a projective variety X ⊂ Pn , there exists a
unique saturated radical ideal I such that V(I) = X. Under mild hypotheses, any
ideal generated by straightening relations on a doset is saturated and radical.
Theorem 3.1. Let D be a doset whose underlying poset has a unique minimal element
α0 , and let A := C[D]/J be an algebra with straightening law on D, where J is the
ideal generated by the straightening relations. Then J is saturated.
Proof. Let f =
Pk
i=1
ai mi 6∈ J be a non-trivial linear combination of (distinct) stan-
dard monomials (i.e., for each i, the support of mi is a chain in D). For each
Pk
N
N ∈ N, pN
α0 f =
i=1 ai pα0 mi is a linear combination of standard monomials, since
N
supp mi ∪ {α0 } is a chain for each i ∈ [k]. It is non-trivial since pN
α0 m i = p α0 m j
implies i = j. Thus pN
α0 f 6∈ J for all n ∈ N, so that f 6∈ sat J.
Theorem 3.2. An algebra with straightening law on a doset is reduced.
Proof. Let A be an algebra with straightening law. For f ∈ A and α ∈ P, denote by
fα the restriction of f to the dual Schubert variety X α .
We will show by induction on the poset P and on n ∈ N that fαn = 0 implies
fα = 0.
Let f ∈ A such that fαn = 0 for some n ∈ N. Since fβn = 0 for all β ≥ α, fβ = 0
by induction. It follows that fα is supported on monomials on X α which vanish on
41
X β for all β ≥ α. That is,
fα =
m
X
i=1
ci peαi p(α,β1,i ) · · · p(α,βℓi ,i ) .
(3.3)
For the right hand side of (3.3) to be standard, we must have ℓi = 1 for all i = 1, . . . m.
Also, homogeneity implies that e := e1 = · · · = em for i = 1, . . . m. Thus, if we set
β1,i := βi , then fα has the form
fα =
peα
m
X
ci p(α,βi ) .
(3.4)
i=1
n
By induction on n we may assume that fα2 = 0 (since (fα2 )⌈ 2 ⌉ = 0).
Choose a linear extension of D as follows. Begin with a linear extension of P ⊂ D.
For incomparable elements (α, β), (γ, δ) of D, set (α, β) ≤ (γ, δ) if β < δ or β = δ
and α ≤ γ. With respect to the resulting linear ordering of the variables, take the
lexicographic term order on monomials in A.
For an element g ∈ A, denote by lt(g) (respectively, lm(g)) the lead term (respectively, lead monomial) of g. Reordering the terms in (3.4) if necessary, we may
assume that lt(fα ) = c1 peα p(α,β1 ) .
Writing fα2 as a linear combination of standard monomials (by first expanding the
square of the right hand side of (3.4) and then applying the straightening relations),
we see that lt(fα2 ) = ±c21 p2e+1
pβ1 . This follows from our choice of term order and the
α
fourth condition in Definition 2.45.
We claim that lt(fα2 ) cannot be canceled in the expression for fα2 as a sum of
standard monomials. Indeed, suppose there are i, j ∈ [m] such that lm((peα p(α,βi ) ) ·
(peα p(α,βj ) )) = p2e+1
pβ1 . Then by the straightening relations, β1 ≤ βi , βj . But β1 6< βi
α
since lm(fα ) = peα p(α,β1 ) . For the same reasons, β1 6< βj . Therefore βi = βj = β1 , so
c1 peα p(α,β1 ) is the only term of fα contributing to the monomial p2e+1
pβ1 in fα2 .
α
42
B. Drinfel’d flag varieties
1. A basis for Sd C2 ⊗ L(ωn )∗
Recall that the Drinfel’d Lagrangian Grassmannian embeds in P(Sd C2 ⊗ L(ωn )). We
describe bases for the representation L(ωn ) and its dual L(ωn )∗ .
¡ ¢
(a)
and positive integers, set vα := sa td−a ⊗ eα1 ∧ · · · ∧ eαn ∈ Sd C2 ⊗
For α ∈ hni
n
V
Vn 2n
∗
(a)
C , and let pα := sa td−a ⊗ e∗α1 ∧ · · · ∧ e∗αn ∈ Sd C2 ⊗ n C2n be the Plücker
coordinate indexed by α(a) ∈ Dd,n .
V
∗
The representation L(ωn )∗ is the quotient of n C2n by the linear subspace
V
∗
Ln = Ω∧ n−2 C2n described in Proposition 2.31. Thus Sd C2 ⊗L(ωn )∗ is the quotient
V
∗
of Sd C2 ⊗ n C2n by the linear subspace:
Ld,n := Sd C2 ⊗ Ln
Note that Ld,n is spanned by linear forms
ℓ(a)
:= sa td−a ⊗
α
for α ∈
¡ hni ¢
n−2
X
i|{ı̄,i}∩α=∅
e∗ı̄ ∧ e∗i ∧ e∗α1 ∧ · · · ∧ e∗αn−2
(3.5)
and a = 0, . . . , d. The linear form (3.5) is simply sa td−a tensored with a
linear form generating Ln (Proposition 2.31). Each term in the linear form (3.5) is a
Plücker coordinate indexed by a sequence (α1 < · · · < ı̄ < · · · < i < · · · < αn−2 )(a) ∈
¡hni¢
, for some i ∈ [n].
n d
Let T ′ ⊂ SL2 (C) and T ⊂ Sp2n (C) be maximal tori. The maximal torus T ′ is
one dimensional, so that its Lie algebra t′ has basis consisting of a single element
H ∈ t′ . Let hi := Eii − Eı̄ı̄ for i ∈ hni, and recall our convention that ¯ı̄ = i, so that
hı̄ = −hi . The Lie algebra t of T ⊂ Sp2n (C) has a basis {hi | i ∈ [n]}. The weights of
the maximal torus T ′ × T ⊂ SL2 (C) × Sp2n (C) are elements of t′∗ ⊕ t∗ . Each Plücker
V
P
(a)
coordinate pα ∈ Sd C2 ⊗( n C2n )∗ is a weight vector of weight (d−2a)H ∗ + i|ᾱi 6∈α h∗αi .
43
Each linear form (3.5) lies in a unique weight space, so we may restrict our
(a)
(b)
attention to one of them. Note that the linear form ℓα is nearly identical to ℓα ; one
simply changes the superscript of each Plücker coordinate from (a) to (b). We may
(0)
(0)
therefore assume a = 0 and write ℓα for ℓα and pα for pα .
¡ hni ¢
, we have an element ℓα = Ω ∧ pα ∈ Ln . This is a weight
For each α ∈ n−2
vector of weight ωα := h∗α1 + · · · +h∗αk ∈ t∗ . Set α
e := {i ∈ α | ı̄ 6∈ α} and observe that
¡ hni ¢
ωαe = ωα . The elements α ∈ n−2 such that ℓα ∈ (Ln )ω are those satisfying ωα = ω.
That is, (Ln )ω = hΩ ∧ pα | ωα = ωi.
The shape of the linear form ℓα is determined by the number of pairs {ı̄, i} ⊂ α,
i.e., the cardinality of α \ α
e; it is the same, up to multiplication of some variables
by −1, as the linear form ℓα\eα = Ω ∧ pα\eα ∈ Ln−2|eα| , of weight ωα\eα . It follows that
the generators of (Ln )ωα have the same form as those of (Ln−2|eα| )ωα , up to some signs
arising from sorting the elements of α and the indices appearing in Ω. Since these
signs do not affect linear independence, it suffices to find a basis for (Ln )0 ; it is then
straightforward to obtain a basis for (Ln )ωα . We thus assume that the weight space
in question is (Ln )0 . This implies that n is even; set 2m := n.
Example 3.6. We consider linear forms which span (L6 )h∗1 +h∗3 . Let m = 3 (so n = 6)
and ω = h∗1 + h∗3 . If α = 6̄136, then α
e = 13 and ωα = ω. We have ℓα = p6̄5̄1356 +
p6̄4̄1346 − p6̄2̄1236 . The equations for the weight space (Ln )ω are
ℓ6̄136 = p6̄5̄1356 + p6̄4̄1346 − p6̄2̄1236
ℓ5̄135 = p6̄5̄1356 + p5̄4̄1345 − p5̄2̄1235
ℓ4̄134 = p6̄4̄1346 + p5̄4̄1345 − p4̄2̄1234
ℓ2̄123 = p6̄2̄1236 + p5̄2̄1235 + p4̄2̄1234
We can obtain the linear forms which span (L4 )0 (see Example 2.39) by first removing
44
every occurrence of 1 and 3 in the subscripts above and then flattening the remaining
indices. That is, we apply the following replacement (and similarly for the negative
indices): 6 7→ 4, 5 7→ 3, 4 7→ 2, and 2 7→ 1. We then replace a variable by its negative
if 2 appears in its index; this is to keep track of the sign of the permutation sorting
the sequence (ı̄, i, α1 , . . . , αn−2 ) in each term of ℓα (see Equation 3.5). To see why this
is necessary, observe that the forms (3.5) are just Ω ∧ e∗α1 ∧ · · · ∧ e∗αn−2 .
From Proposition 2.31, it follows that the map
à 2m
!
Ã2m−2
!
^
^
C4m
C4m
−→
0
0
V
given by contraction with the form Ω ∈ 2 C4m is surjective, with kernel (L(ω2m ))0 .
¡ ¢
V
} is a basis of ( 2k C4m )0 (for any k ≤ m), we have
Since the set {(ᾱ, α) | α ∈ [2m]
k
dim(L(ω2m ))0
2m
2m−2
^
^
4m
= dim( C )0 − dim(
C4m )0
µ ¶ µ
¶
2m
2m
=
−
m
m−2
µ ¶
2m
1
.
=
m+1 m
This number is equal to the number of admissible pairs of weight 0.
Lemma 3.7. dim(L(ωn ))0 is the number of admissible pairs (α, β) ∈ Dn of weight
ωα +ωβ
2
= 0.
Proof. Recall that each trivial admissible pair
α = [ā1 , . . . , ās , b1 , . . . , bn−s ] ∈ Dn
indexes a weight vector, of weight
Pn−s
i=1
h∗bi −
Ps
i=1
h∗ai . Also, the non-trivial admissible
pairs are those (α, β) for which α < β have the same number of negative elements.
From this it follows that the admissible pairs of weight zero are the (α, β) ∈ Dn
45
such that β = [ām , . . . , ā1 , b1 , . . . , bm ], α = [b̄m , . . . , b̄1 , a1 , . . . , am ], {a1 , . . . , am } and
{b1 , . . . , bm } are disjoint. This last condition is equivalent to ai > bi for all i ∈ [m].
The number of such pairs is equal to the number of standard tableaux of shape (m2 )
(that is, a rectangular box with 2 rows and m columns) with entries in [2m]. By the
¡2m¢
1
hook length formula [12] this number is m+1
.
m
V
∗
The weight vectors pα ∈ ( n C4m )0 are indexed by sequences of the form
α = (ᾱm , . . . , ᾱ1 , α1 , . . . , αm )
which can be abbreviated by the positive subsequence α+ := (α1 , . . . , αm ) ∈
¡[2m]¢
m
without ambiguity. We take these as an indexing set for the variables appearing in
the linear forms (3.5).
With this notation, the positive parts of Northeast sequences are characterized
in Proposition 3.9. The proof requires the following definition.
Definition 3.8. A tableau is a partition whose boxes are filled with integers from the
set [n], for some n ∈ N. A tableau is standard if the entries strictly increase from left
to right and top to bottom.
Proposition 3.9. Let α ∈
¡h2mi¢
be a Northeast sequence. Then the positive part of
¡[2m]¢
α satisfies α+ ≥ 24 · · · (2m) ∈ m . In particular, no Northeast sequence contains
2m
1 ∈ [2m] and every Northeast sequence contains 2m ∈ [2m].
Proof. α+ ≥ 24 · · · (2m) if and only if the tableau of shape (m2 ) whose first row is
filled with the sequence (αt )+ = [n] \ α+ and whose second row is filled with the α+
is standard. This is equivalent to α being Northeast.
It follows from Proposition 2.38 that the set N E of Northeast sequences indexing
vectors of weight zero has cardinality equal to the dimension of the zero-weight space
46
of the representation L(ω2m )∗ . This weight space is the cokernel of the map
Ω∧•:
µ2m−2
^
4m
C
¶∗
0
−→
µ^
2m
4m
C
¶∗
.
0
Similarly, the weight space L(ω2m )0 is the kernel of the dual map
Ωy• :
µ^
2m
4m
C
¶
0
−→
µ2m−2
^
4m
C
¶
.
0
¡ ¢
) of
We fix the positive integer m, and consider only the positive part (in [2m]
m
¢
¡
such that the associated Plücker coordinate has weight zero. For
elements of h4mi
2m
¡ ¢
, a matching is a bijection M : α → αc . For any matching, we have an
α ∈ [2m]
m
¡ ¢
element of the kernel L(ω2m ) of Ωy•: Let Hα be the set of all sequences in [2m]
m
obtained by interchanging M (αi ) and αi , for i ∈ I, I ⊂ [m]. Elements of the set Hα
are the vertices of a hypercube, whose edges connect pairs of sequences which are
related by the interchange of a single element. Equivalently, a pair of sequences are
connected by an edge if they share a subsequence of size m−1. Observe that for a
subsequence β ⊂ α of length m−1, there exists a unique edge of Hα connecting two
vertices with common subsequence β. Let I · α denote the element of Hα obtained
from α by the interchange of M (αi ) and αi for i ∈ I. The element
Kα :=
X
(−1)|I| vI·α
I⊂[m]
lies in L(ω2m ). Indeed, for each I ⊂ [m], we have
ΩyvI·α =
m
X
v(I·α)\{(I·α)i } .
i=1
For each term (−1)|I| v(I·α)\{(I·α)i } on the right-hand side, suppose that j ∈ [m] is such
that either (I ·α)i = αj or (I ·α)i = αjc . Set J := I ∪{j} if (I ·α)i = αj and J := I \{j}
if (I · α)i = αjc . The set J is the unique subset of [m] such that (I · α) \ {(I · α)i } is in
47
the support of ΩyvJ·α , with coefficient (−1)|J| = (−1)|I|+1 . Hence these terms cancel,
¡ [2m] ¢
with vβ in the support of ΩyKα is
and we see that the coefficient of each β ∈ m−1
zero. Therefore ΩyKα = 0. See Example 3.13 for the case m = 2.
If α ∈ N E then that there exists a descending matching, that is, M (αi ) < αi for
all i ∈ [m]. For example, the condition that the matching M (αi ) := αic be descending
is equivalent to the condition that α be Northeast. If we choose a descending matching
for each α ∈ N E, the element Kα ∈ L(ω2m ) involves only sequences which preceed α.
It follows that the set B := {Kα ∈ L(ω2m ) | α ∈ N E} is a basis for L(ω2m ).
Lemma 3.10. The Plücker coordinates pα with α ∈ N E are a basis for L(ω2m )∗ .
Proof. Fix a basis B of L(ω2m ) obtained from descending matchings of each Northeast sequence with its dual. We can use this basis to show that the set of Plücker
coordinates pα such that α is Northeast is a basis for the dual L(ω2m )∗ .
P
Suppose not. Then there exists a linear form ℓ = α∈N E cα pα vanishing on each
element of the basis B. We show by induction on N E that all of the coefficients cα
appearing in this form vanish.
Fix a Northeast sequence α ∈ N E, and assume that cβ = 0 for all Northeast
β < α. Since Kα involves only the basis vectors vβ with β ≤ α, we have ℓ(Kα ) = cα ,
hence cα = 0. This completes the inductive step of the proof.
The initial step of the induction is simply the inductive step applied to the unique
minimal Northeast sequence α = 24 · · · (2m).
It follows that every Plücker coordinate pα indexed by a non-Northeast sequence
α can be written uniquely as a linear combination of Plücker coordinates indexed
by Northeast sequences. We can be more precise about the form of these linear
combinations. Recall that each fiber of the map π2m contains a unique Northeast
sequence. For a sequence α0 , let α be the Northeast sequence in the same fiber as α0 .
48
Lemma 3.11. For each non-Northeast sequence α0 , let ℓ′α0 be the linear relation
among the Plücker coordinates expressing pα0 as a linear combination of the pβ with
β Northeast. Then pα appears in ℓ′α0 with coefficient (−1)|I| , where α = I · α0 , and
every other Northeast β with pβ in the support of ℓ′α0 satisfies β > α.
Proof. Let M be the descending matching of α with αc defined by M (αi ) := αic . Let
Kα be the kernel element obtained by the process described above. Any linear form
ℓ = pα0 + (−1)|I|+1 pα +
X
cβ pβ
α<β∈N E
vanishes on Kα .
We extend this relation to one which vanishes on all of L(ωn )0 , proceeding inductively on the poset of Northeast sequences greater than or equal to α. Suppose
that β > α is Northeast. By induction, suppose that for each Northeast sequence γ
in the interval [α, β] the coefficient cγ of ℓ has been determined in such a way that
ℓ(Kγ ) = 0.
Let S be the set of Northeast sequences γ in the open interval (α, β) such that
vγ appears in Kβ . Then
ℓ(Kβ ) =
so setting cβ := −
P
µX
γ∈S
γ∈S
cγ
¶
+ cβ ,
cγ implies that ℓ(Kβ ) = 0.
This completes the inductive part of the proof. We now have a linear form ℓ
vanishing on L(ωn )0 which expresses pα0 as a linear combination of Plücker coordinates
indexed by Northeast sequences. Since such a linear form is unique, ℓ = ℓ′α0 .
By Lemmas 3.10 and 3.11 and the argument preceding them, we have the following theorem.
(a)
Theorem 3.12. The system of linear relations {ℓα
= sa td−a ⊗ Ω ∧ pα | a =
49
¡ hni ¢
} has a reduced normal form consisting of linear forms express¡ ¢
(b)
as a linear combination of
ing each Plücker coordinate pβ with β 6∈ N E ⊂ hni
n
¡hni¢
Plücker coordinates indexed by Northeast elements of n .
0, . . . , d, α ∈
n−2
Proof. We have seen that the linear relations preserve weight spaces, and Lemmas 3.10
and 3.11 provide the required normal form on each of these. The union of the relations
constitute a normal form for the linear relations generating the entire linear subspace
Ld,n .
Example 3.13. Consider the zero weight space
¡V4
C8
¢
(so that m = 2). This
0
is spanned by the vectors vα := eα1 ∧ eα2 ∧ eα3 ∧ eα4 (with dual basis the Plücker
coordinates pα = vα∗ ), where α ∈ {4̄3̄34, 4̄2̄24, 4̄1̄14, 3̄2̄23, 3̄1̄13, 2̄1̄12}. The Northeast
¡V4 8 ¢
¡V2 8 ¢
sequences are 4̄3̄34 and 4̄2̄24. The kernel of Ωy• :
C 0→
C 0 is spanned
by the vectors
K4̄2̄24 = v4̄2̄24 − v4̄1̄14 − v3̄2̄23 + v3̄1̄13
and
K4̄3̄34 = v4̄3̄34 − v4̄1̄14 − v3̄2̄23 + v2̄1̄12 .
To see this concretely, we compute:
ΩyK4̄2̄24 = v4̄4 + v2̄2 − v4̄4 − v1̄1 − v3̄3 − v2̄2 + v3̄3 + v1̄1
= 0,
and similarly ΩyK4̄3̄34 = 0. The fibers of the map π4 :
¡h4i¢
4
→ D4 are
π4−1 (4̄3̄12, 2̄1̄34) = {4̄3̄34, 2̄1̄12}
and
π4−1 (4̄2̄13, 3̄1̄24) = {4̄2̄24, 4̄1̄14, 3̄2̄23, 3̄1̄13}.
50
The expression for p4̄1̄14 as a linear combination of Plücker coordinates indexed by
Northeast sequences is
ℓ4̄1̄14 = p4̄1̄14 + c4̄2̄24 p4̄2̄24 + c4̄3̄34 p4̄3̄34 ,
for some c4̄2̄24 , c4̄3̄34 ∈ C. Since ℓ4̄1̄14 (K4̄2̄24 ) = c4̄2̄24 − 1, we take c4̄2̄24 = 1. Similarly,
ℓ4̄1̄14 (K4̄3̄34 ) = c4̄3̄34 − 1, so c4̄3̄34 = 1. Hence ℓ4̄1̄14 = p4̄1̄14 + p4̄2̄24 + p4̄3̄34 , which agrees
with (2.41).
2. Proof and consequences of the straightening law
We wish to find generators of (Id,n +Ld,n ) ∩ C[Dd,n ] which express the quotient as
an algebra with straightening law on Dd,n . Such a generating set is automatically
a Gröbner basis with respect to the degree reverse lexicographic term order where
variables are ordered by a refinement of the doset order. We begin with a Gröbner
¡ ¢
,
basis GId,n +Ld,n for Id,n +Ld,n with respect to a similar term order. For α(a) ∈ hni
n d
write α̌(a) := α(a) ∨ (αt )(a) and α̂(a) := α(a) ∧ (αt )(a) , so that πn (α(a) ) = (α̂(a) , α̌(a) ).
Let < be a linear refinement of the partial order on Pd,n satisfying the following
conditions. First, the Northeast sequence is minimal among those in a given fiber
of πn . This is possible since every weight space is supported on a antichain (i.e., no
two elements are comparable). Second, α(a) < β (b) if (α̂(a) , α̌(a) ) is lexicographically
smaller than (β̂ (b) , β̌ (b) ).
Consider the resulting degree reverse lexicographic term order. A reduced Gröbner
basis Gd,n for Id,n +Ld,n with respect to this term order will have standard monomials
indexed by chains (in Pd,n ) of Northeast partitions. While every monomial supported
on a chain of Northeast partitions is standard modulo Id,n , this is not always the
case modulo Id,n +Ld,n . In other words, upon identifying each Northeast partition
appearing in a given monomial with an element of Dd,n , we do not necessarily obtain
51
a monomial supported on a chain in Dd,n . It is thus necessary to identify precisely
¡ ¢
correspond to chains in Dd,n via the map πn .
which Northeast chains in hni
n
(a) (b)
A monomial pα pβ with α(a) < β (b) , (β t )(b) and α(a) , β (b) both Northeast cannot
be reduced modulo GId,n or GLd,n . On the other hand, if α(a) < β (b) (say), but α(a)
and (β t )(b) are incomparable (written α(a) 6∼ (β t )(b) ) then there is a relation in GId,n
(a) (b)
with leading term pα pβ t . It follows that the degree-two standard monomials are
(a) (b)
indexed by Northeast partitions pα pβ with α(a) < β (b) , (β t )(b) .
(a) (b)
Conversely, any monomial pα pβ with α(a) < β (b) , (β t )(b) cannot be the leading
term of any element of GId,n +Ld,n . To see this, observe that GId,n +Ld,n is obtained
by Buchberger’s algorithm applied to GId,n ∪ GLd,n , and we may consider only the
S-polynomials S(f, g) with f ∈ GId,n and g ∈ GLd,n . By Proposition 2.10, any such
S-polynomial reduces to zero, unless in< g divides in< f .
(a)
Let α0 be the partition such that in< f = pα0 (that is, f is the unique expression of
(a)
pα0 as a linear combination of Plücker coordinates indexed by Northeast partitions),
and let α be the unique Northeast partition such that πn (α0 ) = πn (α). By the
(a)
reduced normal form given in Theorem 3.12, S(f, g) is the obtained by replacing pα0
(a)
(a)
with ±pα + ℓ, where ℓ is a linear combination of Plücker coordinates pγ
with γ
Northeast and α+ < γ+ . This latter condition implies that α̂ < γ̂ (also, α̌ > γ̌), and
therefore (α̂, α̌) is lexicographically smaller than (γ̂, γ̌).
Hence the standard monomials with respect to the reduced Gröbner basis GId,n +Ld,n
(a) (b)
are precisely the pα pβ with α(a) < β (b) , (β t )(b) .
Recall that elements of the doset Dd,n are pairs (α, β) of admissible (Defini¡ ¢
tion 2.34) elements of hni
such that (regarded as sequences):
n d
• α<β
• α and β have the same number of negative (or positive) elements
52
Equivalently, regarding α and β as partitions, the elements of Dd,n are pairs (α, β) of
symmetric partitions such that
• α ⊆ β,
• α and β have the same Durfee square,
where the Durfee square of a partition α is the largest square subpartition (pp ) ⊂ α
(for some p ≤ n).
¡ ¢
]/hId,n +Ld,n i is an algebra with straightening law on Dd,n .
Theorem 3.14. C[ hni
n d
β̌
α̌
α̌
β̌
βt
β
α
βt
β
αt
αt
α
β̂
β̂
α̂
α̂
Case 1
Case 2
Fig. 8. The two cases in the proof of fourth condition in Definition 2.45.
53
Proof. Since standard monomials with respect to a Gröbner basis are linearly independent, the arguments above establish the first two conditions in Definition 2.45.
To establish the third condition, note that it suffices to consider the expression for
a degree-2 monomial as a sum of standard monomials. For simplicity, we absorb the
¡ ¢
superscripts into our notation and write α ∈ hni
and similarly for the corresponding
n d
Plücker coordinate. Let
p(α̂,α̌) p(β̂,β̌) =
k
X
cj p(α̂j ,α̌j ) p(β̂j ,β̌j )
j=1
be a reduced expression in GId,n +Ld,n for p(α̂,α̌) p(β̂,β̌) as a sum of standard monomials.
That is, p(α̂,α̌) p(β̂,β̌) is non-standard and p(α̂j ,α̌j ) p(β̂j ,β̌j ) is standard for j = 1, . . . , k,
where α (respectively, β) be the unique Northeast partition such that πn (α) = (α̂, α̌)
(respectively, πn (β) = (β̂, β̌)), and similarly for each αj and βj .
Fix j = 1, . . . , k. The standard monomial p(α̂j ,α̌j ) p(β̂j ,β̌j ) is obtained by the reduction of a standard monomial pγ pδ appearing in an element of the Gröbner basis GId,n .
If γ and δ are both Northeast, then nothing happens, i.e., γ = αj and δ = βj . If γ
is not Northeast, then we rewrite pγ as a linear combination of Plücker coordinates
indexed by Northeast sequences. Lemma 3.11 ensures that the leading term of the
new expression is p(γ̂,γ̌) , and the lower order terms p(ǫ̂,ǫ̌) satisfy ǫ̂ < γ̂. The lexicographic comparison in the third condition of Definition 2.45 terminates with the first
Plücker coordinate. That is, if (α̂j ≤ α̌j ≤ β̂j ≤ β̌j ) is lexicographically smaller than
(α̂ ≤ α̌ ≤ β̂ ≤ β̌), then either α̂j < α̂ or α̂j = α̂ and α̂j < α̂. Therefore the reduction
process applied to pδ does not affect the result, and the third condition is proven.
It remains to prove the fourth condition. Suppose that (α̂, α̌) and (β̂, β̌) are
incomparable elements of Dd,n (α and β Northeast). This means that α is incomparable to either β or β t (possibly both). Without loss of generality, we will deal only
with the more complicated case that α and β t are incomparable. The hypothesis of
54
the fourth condition is that the set {α̂, α̌, β̂, β̌} forms a chain in
¡hni¢
n
d
. Up to inter-
changing the roles of α and β, there are two possible cases: either α̂ < β̂ < α̌ < β̌ or
α̂ < β̂ < β̌ < α̌. We visualize each case in Figure 8.
Suppose α̂ < β̂ < α̌ < β̌. Recall that for any γ0 ∈
¡hni¢
n
d
, with Northeast sequence
γ in the same fiber of πn , the expression for the Plücker coordinate pγ0 as a linear
combination of Plücker coordinates indexed by Northeast sequences is supported on
Plücker coordinates pδ such that δ+ ≥ γ+ , with equality if and only if δ = γ, and the
Plücker coordinate pγ appears with coefficient ±1 (Lemma 3.11).
Upon replacing each Northeast (or Southwest) partition with its associated doset
element using the map πn from Chapter II, Section 1, the first two terms of straightening relation for pα pβ t are
pα pβ t − pα∧β t pα∨β t = pα pβ t − σp((α∧β t )∧ ,(α∧β t )∨ ) p((α∨β t )∧ ,(α∨β t )∨ ) + lower order terms
= σβ p(α̂,α̌) p(β̂,β̌) − σp(α̂,β̂) p(α̌,β̌) + lower order terms ,
where σ = ±1. The second equation is justified by the following computation in the
¡ ¢
.
distributive lattice hni
n d
(α ∧ β t ) ∧ (αt ∧ β) = (α ∧ αt ) ∧ (β ∧ β t )
= α̂ ∧ β̂
= α̂
(α ∧ β t ) ∨ (αt ∧ β) = (α ∨ αt ) ∧ ((α ∨ β) ∧ (β t ∨ αt )) ∧ (β t ∨ β)
= α̌ ∧ (α̂ ∨ (α ∧ β t )∨ ∨ β̂) ∧ β̌
= α̌ ∧ β̂ ∧ β̌
= β̂
55
Similarly, (α ∨ β t ) ∧ (αt ∨ β) = α̌ and (α ∨ β t ) ∨ (αt ∨ β) = β̌.
If α̌ < β̂ < β̌ < α̌, then we have both α 6∼ β and α 6∼ β t . We use the relation for
the incomparable pair α 6∼ β t .
pα pβ t − pα∧β t pα∨β t = pα pβ t − σp((α∧β t )∧ ,(α∧β t )∨ ) p((α∨β t )∧ ,(α∨β t )∨ ) + lower order terms
= σβ p(α̂,α̌) p(β̂,β̌) − σp(α̂,β̂) p(β̌,α̌) + lower order terms ,
where the second equality holds by a similar computation in
complete.
¡hni¢
n
d
. The proof is
The next result shows that the algebra with straightening law just constructed
is indeed the coordinate ring of LQd (n).
¡ ¢
Theorem 3.15. C[ hni
]/hId,n +Ld,n i ∼
= C[LQd (n)].
n d
Proof. Let I ′ := I(LQd (n)). Clearly, we have Id,n +Ld,n ⊂ I ′ . Since the degree and
codimension of these ideals are equal, I ′ is nilpotent modulo Id,n +Ld,n . On the other
hand Id,n +Ld,n is radical, so Id,n +Ld,n = I ′ .
Corollary 3.16. The coordinate ring of any Schubert subvariety of LQd (n) is an
algebra with straightening law on a doset, hence Cohen-Macaulay and Koszul.
Proof. For α(a) ∈ Dd,n , the Schubert variety Xα(a) is defined by the vanishing of the
(b)
Plücker coordinates p(β,γ) for γ (b) 6≤ α(a) . The four conditions in Definition 2.45 are
stable upon setting these variables to zero, so we obtain an algebra with straightening
law on the doset {(β, γ)(b) ∈ Dd,n | γ (b) ≤ α(a) }.
Let D ⊂ P ×P be a doset on the poset P, A any algebra with straightening law on
D, and C{P} the unique discrete algebra with straightening law on P. That is C{P}
has algebra generators corresponding to the elements of P, and the straightening
56
relations are αβ = 0 if α and β are incomparable elements of P. Then A is CohenMacaulay if and only if C{P} is Cohen-Macaulay [9].
On the other hand, C{P} is the face ring of the order complex of P. The order
complex of a locally upper semimodular poset is shellable. The face ring of a shellable
simplicial complex is Cohen-Macaulay [4]. By Proposition 2.53, any interval in the
poset Pd,n is a distributive lattice, hence locally upper semimodular. This proves
that C[LQd (n)] is Cohen-Macaulay. The Koszul property is a consequence of the
quadratic Gröbner basis consisting of the straightening relations.
3. Representation-theoretic interpretation
Let G be SL(V ) (dim V = n+1) or Sp(V ) (dim V = 2n) and L(ω) the irreducible
representation with highest weight ω, where ω := ωk is the k th fundamental weight
(for G = Sp(V ), we assume that k = n). The group SL2 (C) × G acts diagonally
on S 2 (S d C2 ⊗ L(ω)), the degree-2 component of the homogeneous coordinate ring
S • (S d C2 ⊗V ). The Drinfel’d Grassmannian Qd (G/Pk ) is invariant under SL2 (C) ×G,
and so S 2 (S d C2 ⊗V ) decomposes as a sum of the degree-2 components A2 and I2 of the
coordinate ring and the defining ideal, respectively. Results in [41] and Theorem 3.14
imply that I2 generates the full ideal I(Qd (G/Pk )).
We answer the following question:
Question 3.17. How do A2 and I2 decompose into irreducible modules for the action
of the semisimple group SL2 (C) × G?
We use the properties of Schur modules [13, 43], which are the irreducible repreP
sentations of SL(V ). For a weight ω = ni=1 ai ωi we have
L(ω) ∼
= Lλ V
57
where λ = (nan , . . . , 1a1 ) and Lλ V denotes the Schur module associated to λ. Our
arguments apply to the case G = Sp(V ) by virtue of Lemma 3.18 below. Suppose
that the partition λ = (λ1 , λ2 ) has two rows. Recall from Chapter II that the Schur
module associated to a partition λ is defined as the image of the map
′
′
φλ : ∧λ1 V ⊗ ∧λ2 V → S λ1 V ⊗ · · · ⊗ S λr V
defined as the composition of the exterior diagonal
δ : ∧λ1 V ⊗ ∧λ2 V →
O
Vi,j
i,j
followed by the multiplication map
m:
O
i,j
On
N
i,j
′
′
Vi,j → S λ1 V ⊗ · · · ⊗ S λr V
Vi,j we have homomorphisms τk,ℓ : V ⊗N → V ⊗(N −2) where N := |λ|,
given by contraction with the symplectic form Ω:
τk,ℓ (v1 ⊗ · · · ⊗ vN ) := Ω(vk , vℓ )v1 ⊗ · · · ⊗ v̂k ⊗ · · · ⊗ v̂ℓ ⊗ · · · ⊗ vN
Lemma 3.18. Let Lhλi V denote the irreducible representation of Sp(V ) with highest
weight λ.
′
′
1. Lhλi V is the image in S λ1 V ⊗ · · · ⊗ S λr V of the intersection of im(δ) with the
kernels of all contraction maps V ⊗N → V ⊗(N −2) .
2. Let K :=
T
T
(
k=1,2
i,j∈[λk ] ker τ(1,i),(1,j) ) be the intersection of the kernels of con-
tractions involving two boxes in the same row. If i ∈ [λ1 ], j ∈ [λ2 ], and i 6= j,
then K ∩ im(δ) ⊂ ker τ(1,i),(2,j) + δ(ker φλ ). Thus Lhλi V = im(K ∩ im(δ)).
3. Lhλi V has a basis consisting of standard tableaux of shape λ such that each row
is filled with an admissible pair of sequences.
58
Proof. Statement (1) is well-known; see [13, Section 17.3]. For Statement (2), let
T ∈ K ∩ im δ. In particular, T ∈ ker τ(1,i),(1,j) . Let σ : V ⊗N → V ⊗N be the
map exchanging the (1, j)- and (2, j)-entries. Then m(T ) = m(σ(T )), and σ(T ) ∈
ker(τ(1,i),(2,j) ). Thus T − σ(T ) ∈ δ(ker φλ ), hence T ∈ ker τ(1,i),(1,j) +δ(ker φλ ). The
third statement is a consequence of the second and the combinatorics of bases of
fundamental representations of Sp(V ).
The next step is to determine how S 2 (S d C2 ⊗ L(ω)) itself decomposes. Define
the partition θp := (k+p, k−p) for p = 1, . . . , min{k, n−k} and θp := ∅ for all other
integer values of p, with the convention that V∅ = 0. Also, for Sp(V ) we adopt the
convention that L(n+p) V = L(n−p) V .
Proposition 3.19. The SL2 (C) × G-module S 2 (S d C2 ⊗ L(ω)) is isomorphic to
!
! Ã
Ã
MM
MM
S 2d−4l−2 C2 ⊗ Lθ2m+1 V
S 2d−4l C2 ⊗ Lθ2m V ⊕
l≥0 m≥0
l≥0 m≥0
.
Proof. We have
S 2 (S d C2 ⊗ L(ω)) ∼
=
¡
¢ ¡
¢
S 2 (S d C2 ) ⊗ S 2 L(ω) ⊕ ∧2 (S d C2 ) ⊗ ∧2 L(ω) .
Each module on the right decomposes further:
S 2 (S d C2 ) ∼
=
∧2 (S d C2 ) ∼
=
S L(ω) ∼
=
2
M
S 2d−4k C2 ,
k≥0
M
S 2d−4k−2 C2 ,
k≥0
M
m≥0
Lθ2m V,
59
and
∧2 L(ω) ∼
=
M
Lθ2m+1 V.
m≥0
The first two isomorphisms are as SL2 (C)-modules and the last two are as G-modules [13,
Exercise 15.32]. Since the action of SL2 (C) × G is diagonal, the resulting direct sum
is an irreducible decomposition into SL2 (C) × G-modules.
These modules can be visualized as in Figure 9. The shaded boxes correspond to
the modules appearing in Proposition 3.19. We will see that the irreducible modules
belonging to A2 are those corresponding to the shaded boxes above the diagonal line
(Theorem 3.22).
2d−12
2d−10
2d−8
2d−6
2d−4
2d−2
2d
θ0 θ1 θ2 θ3 θ4 θ5 θ6
Fig. 9. The module S 2 (S d C2 ⊗ L(ω)) decomposes into factors corresponding to the
colored boxes.
60
For a two rowed partition λ and p ≤ λ1 −λ2 , define λ(p) := (λ1 +p, λ2 ). If
µ = (µ1 , µ2 ) ⊂ λ = (λ1 , λ2 ), define λ/µ(p) := λ(p)/µ(p). Set κ(p) := (k + p, k)/(p, 0).
Theorem 3.14 and the results of [41] imply that A2 is spanned by the monomials
(a) (b)
pα pβ , where the tableau obtained by filling the shape κ(b−a) with the entries in α
(in the first row) and β (in the second row) is standard.
Theorem 3.20. The G-module A2 is isomorphic to
d
M
¡
¢⊕(d−p)
Lκ(p) V
p=0
where Lκ(p) V denotes the Schur module corresponding to the skew Young diagram
κ(p) := (k + p, k)/(p, 0).
The connection between the representations appearing in Proposition 3.19 and
Theorem 3.20 is given by the following lemma [43, p. 78, Exercise 3].
Lemma 3.21. Let µ ⊂ λ be two-rowed partitions.
1. There exists a surjective G-morphism
ψ : Lλ/µ V → Lλ/µ(−1) V
2. ker ψ ∼
= L(λ1 −µ2 ,λ2 −µ1 ) V
Proof. The module Lλ/µ V is spanned by fillings of λ/µ with vectors in V . The map
ψ is defined in terms of this basis by shifting the first row of a tableau one unit to the
left. This map is surjective, and the kernel is the preimage of the submodule spanned
by sums of the form
X
u+µ1 −µ2
1 −µ2 +1
σ∈Sλ
(−1)ℓ(σ) U1 ∧ · · · ∧ Uu ∧ Wσ(1) ∧ · · · ∧ Wσ(u−µ1 +1)
⊗ Wσ(u−µ1 +2) ∧ · · · ∧ Wσ(λ1 −µ2 +1) ∧ V1 ∧ · · · ∧ Vv
61
where u+v = λ2 −µ1 −1 and Ui , Vj , Wk are elements of V for i = 1, . . . , u, j = 1, . . . , v,
k = 1, . . . , λ1 −µ2 +1 and Srs denotes the set of permutations σ on s letters such that
σ(1) < · · · < σ(r) and σ(r+1) < · · · < σ(s). These sums over shuffles can be
thought of graphically in terms of a Young scheme, in which the vectors U1 , . . . , Uu
and V1 , . . . , Vv are positioned in the appropriately labeled boxes, and the vectors
W1 , . . . , Wλ1 −µ2 +1 are shuffled (with signs) among the boxes labeled with a •. See
Figures 10 and 11.
Thus, ker ψ consists of linear combinations of tableaux corresponding to the
Young schemes in which the shuffled entries in the two rows just fail to overlap.
U
V
Fig. 10. A Young scheme on λ/µ(−1)
U
V
Fig. 11. A Young scheme on λ/µ
62
Any such linear combination lies in the span of the ordinary (overlapping) Young
schemes and the non-overlapping Young scheme obtained by taking u = 0. The former
vanish on Lλ/µ(−1) V , and the latter is equivalent to the condition that the resulting
linear combination is alternating in the vectors U1 , . . . , Uu , W1 , . . . , Wλ1 −µ2 +1 . Therefore we obtain a (G-linear) isomorphism
ker ψ → L(λ1 −µ2 ,λ2 −µ1 ) V
by sending the sum over shuffles of a tableau on λ/µ to the standard tableau on
(λ1 −µ2 , λ2 −µ1 ) given by putting the vectors U1 , . . . , Uu , W1 , . . . , Wλ1 −µ2 +1 in order
on the first row.
It follows that
Lκ(p) V ∼
=
p
M
Lθm V.
m=0
Using Proposition 3.19, we identify which modules appear in the irreducible decomposition of A2 . This is a sum over m of some SL2 (C) × G-modules M ⊗ Lθm V . These
come from the modules Lκ(p) V for p = m, . . . , d. The SL2 (C)-weights appearing in
these modules are of the form 2d−2p−4q for p = m, . . . , d, q = 0, . . . , d−m. The
(a) (b)
parameter q corresponds to the d−m+1 copies of Lκ(p) V spanned by pα pβ (with
p = b−a ≥ 0) such that the tableau obtained by filling the skew partition κ(p) with
the entries of α in the first row and β in the second row is standard. Thus, for
p−m ≡ 0 mod 2, the contributions from Lκ(p) and Lκ(p+1) are exactly the weights of
S 2d−2m−4p C2 , so the modules S 2d−2m−4p C2 ⊗ Lθm V are irreducible components of A2 .
Summing over m = 0, . . . , d, we have:
Theorem 3.22. The irreducible decomposition of A2 as an SL2 (C) × G-module is
d−m
A2 ∼
=
⌋
d ⌊M
2
M
m=0 p=0
S 2d−2m−4p C2 ⊗ Lθm V
63
The summands of A2 described in Theorem 3.22 are those corresponding to the
boxes above the diagonal line in Figure 9. It follows that the defining ideal is generated
by the direct sum of modules corresponding to the boxes below the line.
64
CHAPTER IV
SUMMARY
The Grassmannian and Lagrangian Grassmannian are important test cases for understanding the flag varieties. Indeed, these were among the first spaces to which
standard monomial theory was successfully applied [38, 30, 28, 29, 31]. While flag
varieties provided the initial motivation, standard monomial theory has been applied
to the closely related Bott-Samelson varieties, which give a resolution of singularities
of Schubert varieties [26, 27, 25]. As the Drinfel’d flag varieties are singular, this suggests that one might hope to find a resolution of its singularities which also admits a
straightening law.
The current situation with respect to the Drinfel’d flag varieties is similar to the
early stages of the development of classical standard monomial theory. Taken together
with the results of Sottile and Sturmfels [41], our results suggest that similar results
might be obtained for other Drinfel’d flag varieties. The work of Littelmann [33] provided the tools needed for a fully general standard monomial theory for flag varieties
(avoiding all considerations of special cases) [24]. It is interesting to ask if such an
approach can be applied to the Drinfel’d flag varieties. One motivation for this would
be a new geometric derivation of intersection numbers in quantum cohomology.
General results of algebras with straightening law can be found in [3, 7, 8, 5,
9]. The results presented in Chapter III, Section A represent some progress in this
direction, and more might yet be done. For example, any Schubert variety of the
Drinfel’d Grassmannian admits a deformation to a toric variety [41], which implies
that the Schubert variety is normal and has rational singularities. In some special
cases, similar results exist for Schubert varieties of a flag variety [8, 15, 34]. While it is
highly desirable to obtain a similar result for the Drinfel’d Lagrangian Grassmannian
65
(or other Drinfel’d flag varieties), it is also interesting in general.
Question 4.1. Under what conditions does there exist a toric deformation of an
algebra with straightening law?
The algebras with straightening law are a interesting class of finitely generated
commutative algebras, and arise in vital areas of algebraic geometry. Answers to
questions such as Question 4.1 would have important consequences; for example,
that the associated varieties are normal with rational singularities.
66
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71
VITA
Name:
James Vincent Ruffo
Address:
Department of Mathematics
Mailstop 3368
Texas A&M University
College Station, TX 77840
Email Address: jruffo@math.tamu.edu
Education:
B.A., Mathematics, University of Rochester, 1996
B.S., Physics, University of Rochester, 1996
M.A., Mathematics, University of Massachusetts-Amherst, 2004